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MECHANISM 



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CONTENTS. 



CHAPTER PAGE 

I. INTRODUCTORY . . . ' i 

II. ON THE CONVERSION OF CIRCULAR INTO RECIPROCATING 

MOTION . . . . , . . . . . 42 

III. ON LINKWORK. . no 

IV. ON THE CONVERSION OF RECIPROCATING INTO CIRCULAR 

MOTION . 148 

> V. ON THE TEETH OF WHEELS 168 

v VI. ON THE USE OF WHEELS IN TRAINS . ... . . 190 

VII. AGGREGATE MOTION . . 215 

VIII. ON TRUTH OF SURFACE AND THE POWER OF MEASUREMENT 266 

j IX. MISCELLANEOUS CONTRIVANCES 284 

H 

APPENDIX . . 341 

INDEX . . . 355 

INDEX TO APPENDIX ........ 360 



713782 

;ineerii 
ibrary 



Engineering 
Lit) 







ELEMENTS OF MECHANISM. 



CHAPTER I. 

INTRODUCTORV. 

A MACHINE is an assemblage of moving parts, constructed for the 
purpose of transmitting motion or force, and of modifying, in 
various ways, the motion or force so transmitted. 

In order to form a definite idea of the meaning which attaches 
to the word ' machine,' it may be useful to refer to an example 
commonly met with such as an ordinary sewing machine. The 
apparatus is rightly called a machine, as being capable of doing 
work of one definite kind, under the simple condition that some 
natural source of energy shall bear upon it and set the working 
parts in motion. Upon looking into its construction we should 
find a fixed framework supporting combinations of movable parts, 
whereof some are employed in actuating a needle and shuttle, 
while others carry forward the material which is to be stitched. 
The movable parts are constrained to take certain definite motions, 
which are arranged beforehand, while some natural force, such as 
the power of the hand or the foot, is applied to the proper recipient, 
and then the machine does work as a necessary consequence of 
the action of the motive power. 

In commencing the systematic study of machinery, it will be 
readily understood that certain simple relations of motion are 
traceable between the prime mover which starts the machinery 
B 



2 Elements of Mechanism. 

and the pieces which execute the work ; and it is also clear that, 
in practice, relations governing the transmission of force must exist 
as certainly as those which govern the transmission of motion 
The considerations relating to force may often occupy the mind 
of the mechanic in a greater degree than those which refer to mo- 
tion; but in reducing the subject to analysis it will be found con- 
venient to separate the two points of view, and to confine our 
attention in the first instance mainly to Theoretical Mechanism 
that is, to an examination of the various contrivances and arrange- 
ments of parts in machinery whereby motion is set up or modified 
and to disregard or postpone any enquiry into the mechanical 
laws which control the forces concerned in these movements. But 
as the present work is intended for use and study by practical men, 
the author will to a small extent break through this general rule, 
and will take occasion, where the enquiry would be useful, of 
pointing out also the manner in which certain pieces of mechanism 
have served a compound object in transmitting exact and definite 
amounts of motion, while dealing at the same time with refined 
and subtle distinctions as to the method of transmitting force. 

We have now to consider and arrange the method according 
to which our enquiries are to be carried on, and if we were to 
pause for a moment and look back upon that rapid creation of 
machinery which followed so closely upon the splendid invention 
of the steam engine by Watt, we should naturally expect that some 
uniform arrangement for applying steam power would be adopted 
by common consent, and that this arrangement would powerfully 
influence the art of constructive mechanism. Accordingly we find 
that in applying the power derived from steam for the purpose of 
driving machinery in our mills and factories, it is the practice to 
connect the engine with a heavy fly wheel, the rotation of which is 
made as uniform as possible, and then to carry on, by lengths of 
shafting, the uniform motion of the fly wheel to each individual 
machine in a factory. 

Suppose, for example, that we were visiting a cotton mill, and 
were examining and endeavouring to comprehend the action of a 
complete piece of machinery, such as a power loom for weaving 
calico. We should at once see that every moving part, acting to 
produce the required result, derived its motion from the uniform 



Introductory. 3 

and constant rotation of a disc or pulley, outside the machine 
itself, and communicating by means of a band with the shafting 
driven by the engine, and thus it would become obvious that the 
problem of making a machine resolved itself mainly into a question 
of the resolution or transfer of circular motion in every variety of 
manner, and subject to every possible modification. 

We shall therefore commence, in the present chapter, with 
some general observations on the conversion and transfer of motion 
in the simple primary forms under which it is to be regarded at 
the outset of the study of mechanism, reserving a more complete 
discussion of the different divisions of the subject for the remaining 
chapters, each of which will treat of movements of a particular 
class. 

It will soon become apparent that, by combining, transferring 
or modifying simple modes of motion, an almost endless variety 
of mechanisms may spring into existence, and our object will be 
to classify and arrange these mechanisms in such a manner that 
the reader may acquire a fair knowledge of what has been already 
accomplished, and may trace the principles which have been de- 
veloped in the construction of many well-known machines. 

To the geometrician a straight line or a plane surface are crea- 
tions of the mind. Euclid, more than 2,000 years ago, had as 
complete a conception of a straight line or a plane as we have at 
the present day, but he could not realise his conceptions even 
approximately, by reason that accuracy of surface was at that time 
a thing unknown. 

And even now how few among young mechanics are aware of 
the exact conditions under which truth of surface has been 
originated, or that a difference of length of 40000 of an inch is 
a quantity which can be palpably and unmistakably measured by 
a workshop instrument, without the aid of a microscope or magni- 
fying lens of any kind. It is not enough to describe machines on 
paper, and to say that they will effect such and such results. The 
science of mechanism is a practical science; it must be more than 
a speculative creation ; the principle of each movement must be 
embodied in shaped pieces of suitable material, and there must be 
some method of testing the exact form or dimensions of the 
several parts. It follows that a knowledge of the steps which have 



4 Elements of Mechanism. 

given to the mechanician the two aids upon which he mainly relies, 
viz. truth of surface and the power of measurement, will form an 
essential portion of the subject-matter upon which we have to 
treat. 

ART. i. To commence with a few enquiries relating to the 
motion of a point in space a point being that ideal thing called 
a material particle, which is defined in a Cambridge text-book as 
being 'a portion of matter indefinitely small in all its dimensions, 
so that its length, breadth, and thickness are less than any assign- 
able linear magnitude ' we shall treat the motion of such a point 
as a simple matter of geometry, all its movements being exact of 
their kind. 

There are three primary cases : 

I. The point may move in a straight line. In such a case the 
direction of its motion remains constant, being that of the line in 
which it moves. 

II. The point may move in one plane, but may continually 
change the direction of its motion. 

III. The point may change its direction so as to move in a 
curved line of any kind. 

In the second case the point is said to move in a plane curve, 
for, according to geometers, a curve is a line traced out by a moving 
point, which is continually changing the direction ot its motion, 
and a plane curve is one which lies in a given plane. 

Conceive that a point is describing a plane curve AB ; then the 
straight line in which the point would 
move at P, if it there ceased to change 
the direction of its motion, is called the 
tangent to the curve AB at the point P 
that is to say, the direction of the 
motion of a moving point is at each 
instant the tangent to its path when the path is a curve. 

It should be borne in mind that we have spoken of a material 
particle as the moving thing, because a material particle has the 
property of mass, and cannot change its velocity or direction 
abruptly unless it be subjected to the action of an infinite force, 
though of course it may be moved abruptly in the sense of being 
set in motion from a position of rest. 




Velocity. 5 

In the third case the point may move in a curve traced upon 
the surface of a cylinder, and not lying in one plane, as, for ex- 
ample, if it followed the outline of the banister of a cylindrical 
staircase. 

The definition of a tangent previously given applies generally 
to all curves, and if a point be moving in a straight line at any 
instant it must also be moving in some definite plane; hence we 
may describe the motion now under consideration by saying that 
the direction of the curve at any instant is continually changing 
by the twisting of the plane in which the point is then moving 
about its tangent line, 

ART. 2. We have next to speak of the velocity of the moving 
point. 

So long as a point is moving continuously, we can form an 
idea of the rate at which it is changing its position relatively to 
other points which are assumed to be at rest. This rate of change 
of position is the velocity of the moving point. 

Velocity may be either uniform or variable. 

The word 'uniform' indicates that the lengths of path de- 
scribed by the moving point in equal times are always the same, 
and the word ' variable ' is applied when the rate of change of 
position of the moving point is continuously altering. The latter 
term would not apply to a step-by-step movement. 

When the velocity of a point is uniform, it is measured by the 
length of path passed over in a unit of time that is, in technical 
language, by the space described in the unit of time. 

The unit of space is usually one foot, and the unit of time is 
one second. 

Hence if a point be moving uniformly with a velocity v, it 
will describe v feet in any second, and will describe tv feet in 
/ seconds. 

Let s be the space described in / seconds, then 

s = tv, or z>=~. 

Def. A foot- second of velocity is the velocity which would cause 
a point to move uniformly through a foot in every second. If the 
velocity of a point be n foot-seconds the point will move uniformly 
through n feet in each second of time. 



6 Elements of Mechanism. 

ART. 3. If the velocity of the point be variable we must look 
to the rate at which s changes as / flows on, or to the ratio be- 
tween the so-called fluxions of s and /. The word ' fluxion ' was 
introduced at the time of Newton. 

Suppose that in time A/ the point describes a space As, and 
that its velocity in the same time increases or decreases continu- 
ously and becomes v + &v. The space it actually describes lies 
between the spaces it would describe if its initial and final velo- 
cities were continued uniform during time A/, or 

vt, AJ, (v 
are in order of magnitude, and so are 



Now let A/, and consequently As, A?; be diminished indefinitely, 

in which case let -- become -,-, then the first and third terms 
A/ dt 

become equal to that lying between them, or v = -/. 

dt 

Hence v = - T for variable motion. 
dt 

In other words, velocity, when uniform, is measured by the 
space described in a unit of time ; when variable it is measured by 
the space which would be described in a unit of time, if the point 
retained throughout that unit the velocity which it has at the in- 
stant considered. The above statements apply equally whether the 
point be moving in a straight line or in a curved line of any kind. 

It is further apparent that the velocity of a point at any instant 
may be represented by a straight line, for the direction of motion 
of the point will be the direction of the line, and the numerical 
measure of the velocity will determine the number of units of 
length in the line. 

Inasmuch as a straight line can be drawn in any direction from 
a point, and since it is usual to describe the straight line as positive 
when it is drawn in one direction from a point, and negative when 
it is drawn from the same point in the opposite direction, so velo- 
cities can be similarly described as positive or negative, according to 
their directions in the same straight line. 



Resolution of Velocities. 




ART. 4. We pass on to the resolution and composition of 
velocities. Conceive that a point is moving uniformly in the 
straight line PQR with a velocity ?', 
and let P, Q, R be three positions of 
the moving point 

Take any two straight lines Ox, 
Qy inclined at a given angle, and 
lying in a plane passing through 
PQR ; draw P/, Q?, Rr respectively 
parallel to Oy ; then pq : pr \ \ PQ 
: PR, and the motion of p along Qx will be in a constant ratio to 
the motion of P along PR. 

The point / is called the projection of P on Ox, and the velo- 
city of/ is said to be the velocity of P resolved along Ox. 

In like manner, if a straight line AR be taken to represent 
the velocity existing in a point at any 
instant, and Ax, Ay be any two straight 
lines intersecting in A, and if the parallelo- 
gram APRQ be completed, the sides AP, 
AQ will represent the resolved velocities 
of the point in directions Ay, Ax. It is 
usual to speak of AP, AQ as the compo- 
nents of the velocity AR ; and, again, AR is called the resultant 
of the component velocities. Hence the following proposition : 

Prop. If there be impressed simultaneously on a particle at 
A two velocities which would separately be represented by the 
adjacent sides AP, AQ (fig. 3) of the parallelogram APRQ, the 
actual velocity of the point will be represented by that diagonal 
AR which passes through the point A. 

Cor. i. Let the angle PAQ be a right angle, and let RAQ=, 



FIG. 3. 




FIG. 4 . 



FIG. 5. 




AR = v. Then will AQ = v cos a, AP = QR = v sin a. 



8 Elements of Mechanism. 

Cor. 2. Let velocities represented by the three sides of a tri- 
angle taken in order viz. AQ, QR, RA be impressed at the same 
instant on a point ; then no motion will ensue, or the point will 
remain at rest ; for the velocities AQ, QR are equivalent to a 
single velocity AR, and the velocities AR, RA are equal and 
opposite, and therefore destroy each other. 

ART. 5. It has been stated that the most simple case of motion 
is that of a point describing a straight line with a uniform velocity, 
also that the motion of a point in a straight line may be the 
aggregate of two movements in lines at right angles to each other, 
and that this is true whether the motion of the point in a straight 
line be uniform or variable. 

But one imperative condition must be observed, viz. that the 
amounts of the corresponding movements in the two perpendicular 
lines must be in a fixed ratio to each other. When this condition 
fails, the point will describe a curve and not a straight line. 

The whole learning of analytical geometry proceeds on the 
doctrine of the composition of motion. If we wish to represent a 
curve by means of a relation between symbols, called an equation 
to the curve, we begin by drawing two straight lines xQx', j'Oy at 
right angles to each other, and employing them as lines of reference. 
For example, let the curve pass 
FlG- 6> through point 6 ; take P any point in 

the curve, and draw PN perpendicular 
to CXr ; let ON=^, NP==>>; then, if y 
be to x in a fixed ratio, the point P 
must of necessity lie in a straight line 
* passing through O. 

Whereas if the ratio between y and 
x varies for every position of the moving 
point P, that point will describe a curve. 
Let P describe a circle whose centre 
is C, and let CO=a ; join CP: then 
NC=OC-ON=a-x 

But CP 2 =PN 2 + NC 2 . 




Circular Motion of a Point. 



FIG. 7. 



The above relation is satisfied only by points lying in the 
circle, and gives an analytical representation of the particular 
curve to which reference has been made. 

The lines x, y are called the co-ordinates of the point P, and 
the axes xOx', yOy' are the axes of co-ordinates ; also the signs 
+ and are employed to indicate the position of P in any par- 
ticular quadrant ; thus if P were situated anywhere in the quad- 
rant x'Oy, the corresponding values of x and y would both be 
negative. 

ART. 6. We are now in a position to discuss the nature of cir- 
cular motion, and may premise that the belief held by the ancients 
with regard to it was fanciful in the extreme, and is obviously 
untenable It was said that the motion of a point in a circle was 
simple, in the sense that it was not made up by putting together 
other separate movements, a doctrine in direct opposition to that 
just laid down. 

The modern belief is that the point P, while describing the arc 
OP of the circle OBDE, may have 
been the subject of two independent 
movements, one from O to N in the 
direction of the diameter OD, and 
the other from N to P in a perpei 
dicular direction. 

Thus let OCD, BCE be two 
diameters of a given circle at right 
angles to each other, P any point in 
the circumference, PN, PM perpen- 
diculars on OC and BC respectively. 

Let P describe the circumference 
with a uniform velocity ; then the 

point N will travel to and fro along OCD with a variable velocity, 
while at the same time the point M moves at a varying, rate up 
and down through BCE. 

The motion of the point N is distinguished by a technical 
name, according to the following definition : 

Def. When a point P moves itniformly in a circle, the ex- 
tremity N of the perpendicular PN let fall from P upon a fixed 
diameter OD has a simple harmonic motion. 




It appears that this is nearly the case with such bodies as the 
satellites of Jupiter when seen from the earth. 

The term ' harmonic motion ' has been chosen for designating 
one component of circular motion because it represents approxi- 
mately the motion of a particle in the various media in which 
waves of sound, light, and heat are propagated. Thus a point at 
the end of the leg of a vibrating tuning-fork has a simple harmonic 
motion very approximately. 

We conclude that circular motion is of a compound character, 
and is capable of resolution into its elements. If it be thus re- 
solved, and if one equivalent be suppressed, so that the motion of 
N is substituted for that of P, we obtain the fundamental case of 
the conversion of circular into straight-line motion. 

And, further, we regard circular motion as compounded of two 
simple harmonic motions in lines at right angles to each other, 
but so related that one component comes to rest when the other is 
in the middle of its swing. 

It will be found that harmonic motions enter into the analysis 
of many forms of mechanism, and that no progress can be made 
without some knowledge of the laws here sketched out. 

The following technical terms are introduced in order to state 
the nature of the movement correctly. 

Def. The amplitude of a simple harmonic motion is half the 
distance between two extreme positions. In other words, it is the 
radius of the auxiliary or bounding" circle. 

Def. The period is the interval of time between two successive 
passages through the same position in the same direction that 
is, it is the time of describing the complete circle. 

Def. The phase is that fraction of the period which has elapsed 
since the moving point was at its extreme position in the positive 
direction. 

In applying these definitions we should say that M and N have 
the same amplitude, that the have the same period, and that they 
differ in phase by of the period ; whence we finally conclude 
that uniform circular motion is compounded of two simple har- 
monic motions of equal period and amplitude, taking place in 
lines at right angles to each other, and differing in phase by one- 
quarter of the whole period. 



Jj tju,^^***^*- K*~*~O^b-tA^ x -t>l>l^/f ^ lislsisi^ **X/1^L^- 

^7^; .Simple Harmonic Mation. 7 1 1^ 

t/(j4yt "** -&**t *W /3 , (<f* 04 t U9 ^^ ^x-<xx^-icxu-w 

/ ART. 7. In order to express the relation between the posi- 
tions P and N we proceed as follows : 

Let CP=, PCO=C. 

Then ON = OC-CN 

= OC - CP cos C. 

Or ON = a (i cos C). 

Ex. To find the position of CP 
when N is half-way between O and C. 




.-.- = (i- cos C), 

whence i cos C=-, cos C=i -=-- 

2 22' 

or C=6o. 

It follows from this that N describes the first half of OC in 
double the time which it takes to describe the other half. 

In further analysing the movement we have to ascertain the 
relation which exists between the actual velocities of P and N at 
any instant. 

Suppose P to sweep round with a uniform velocity, and we 
see at once that N begins to move slowly at O, comes to rest 
gradually at D, and that its greatest speed occurs when just passing 
through C. 

The motion of N is not uniform, and its rate of advance at 
any given instant may be deduced from the so-called triangle of 
velocities. 

Thus take PC, which is perpendicular to the direction of 
motion of P, to represent its velocity in magnitude ; then PN, 
NC represent the components of the velocity of P in directions 
NC, PN respectively. 

, = |N =sinpcN=sinC 

The sine of C has all values from o to i which are registered 
in a table of natural sines, and by substitution from this table we 
can find how much the velocity of P differs from that of N at 
any period of the motion. 



12 Elements of Mechanism. 

When C=45, sin =7071068, 

/. vel. of N= T 7 5 1 7 T7 vel. of P, very nearly. 

When C=9o, sin C=i, .'. vel. of N=vel. of P, 

which is evidently true, because the point P is then moving in a 

direction parallel to OD. 

We observe also that the motion may be divided into four 
equal portions, and that the advance of N from O to C has an 
exact counterpart on the return from D to C, and so for the other 
divisions. 

ART. 8. It happens that the composition of two harmonic 
motions in lines at right angles to each other can be readily verified 
by experiment. For this purpose two tuning-forks, each four or five 
times as large as an ordinary tuning-fork, are required. A small 
polished steel mirror is attached to the outside of the leg of one 
fork in such a manner that the vibration of the leg would cause 
the image of a small luminous spot to describe a straight line. 
The fork is then mounted so that the line in question becomes 
vertical. The second fork is provided also with a small reflecting 
mirror, and is caused to vibrate in a horizontal plane, whereby the 
image of the luminous point, as taken from the first mirror, describes 
a horizontal line when the first fork is at rest. 

We have thus two separate motions in lines at right angles to 
each other, and it has been stated that the motion of a point in the 
leg of a tuning-fork is a simple harmonic motion. 

When the two forks vibrate together, the spot of light describes 
a path which represents the result of compounding the two sepa- 
rate harmonic motions. That path is found to change from a 
straight line to an oval, which presently swells out to a complete 
circle, and then sinks down again to a straight line. The circle 
and the straight line are the two extreme cases between which the 
intermediate curves are to be found. 

The fact is that the forks are not exactly in unison, and that 
one is continually gaining upon the other. If they start together, 
a straight line is the result. If one has gained upon the other so 
far that it is half a swing in advance, the straight line has changed 
into a circle. 

ART. 9. The velocity of N may be found by analysis. Thus 
let ON=*, PCN=0, CP=a. Then x=a (i- cos 0). 



Diagram of Velocities. 



vel. of P= 



dt 



= * , and vel. of N= 
at 



dt 



vel. of P 



dt 



dt 



Another easy proof is the following ; 
Assume that P retains its present ve- 
locity, and let it move in any time to T. 
Draw TN' perpendicular to CN, and PR 
perpendicular to TN', and let PCO-0, 
vel. of N _ NN' 



=* sn , 
at 



FIG. 9. 
T 



then 



PT 




ART. 10. The changes which occur in the velocity of N 
may be set out in a diagram. 

Let C be the centre of the 
circle described by P; take FK, 
AB, two diameters at right angles; 
and draw PM, PN, respectively 
perpendicular to them. Also take 
CS=CM=PN, and join FS. 

Then in the triangles PMC, 
FCS the sides CM, CP are re- 
spectively equal to CS and CF, 
and the angle FCP is common 
to both ; therefore the triangles 
are equal, 

and angle FSC=angle CMP. 

But CMP is a right angle ; therefore CSF is also a right angle, 
and S lies in the circumference of a circle described on CF as a 
diameter. 

It follows that if two circles be described on CF, CK as dia- 
meters, and any straight line CSP be drawn as in the diagram, the 
ratio of CS to CP will be the ratio of the velocities of the points 
N and P respectively. 

At the points F and K we have CS=CP, and the velocities of 
N and P become equal, which is evidently true. v 




Elements of Mechanism. 



ART. ii. The motion of N may be derived from that of P 
by a simple mechanical arrangement. 

Let P represent a small pin set in a circular plate, which is 
movable about C as a centre 
of motion, and let the pin 
work in a groove EF whose 
direction is at right angles to 
that of the sliding bar BN. 
_^_ The bar is, of course, rigidly 
attached to EF, and is con- 




J 



i--- 



FlG. 13. 



strained by guides to move in 
a line pointing to C. 

Of the two equivalents 
which combine to produce the circular motion of P that which 
occurs in the direction FE is rendered inoperative, and the whole 
of the other equivalent is imparted to the sliding bar ; in this way, 
then, the bar moves to and fro as the disc rotates upon its centre, 
and any point in it reproduces exactly the motion of N. 

Referring to the drawing, which is sketched from a model in- 
tended to exhibit simultaneously the motions of the points M and 
N, it will be seen that a pin P passes through the grooved bars 

EF and KL, which over- 
lap each other and are 
connected by slender 
rods sliding between 
guides. Also x and y 
are small balls or index 
fingers, each of which 
traverses over a gradu- 
ated scale, and indicates 
the movement of the cor- 
responding bar. As the 
point P travels round a 
circular groove, it is ap- 
parent that x and y re- 
spectively reproduce the 
movements of N and M. 
ART. 1 2. Having satisfied ourselves as to the nature of circular 




Transfer of Circular Motion. 1 5 

motion, the next step is to consider the manner in which it may 
be modified or transferred. 

There are two fundamental cases : 

1. We may draw the circular motion from the circumference 
of the circle, as we should draw off a piece of string from a reel 
on which it has been wound. 

2. We may suppress one of the two components of the circular 
motion, and may take the other in the form of a complete har- 
monic motion. 

Each method leads to a variety of useful mechanisms. 

As to (i), when two circles touch each other, the direction of 
each curve at the point of contact is that of their common tangent, 
and hence a moving point may be readily transferred from one 
circle into another touching it. 

FIG. 13. 





Thus a point describing the circle A with an uniform velocity 
may be passed into the circle B, and will describe that also with 
'the same linear velocity. 

If the circle B touch the circle A externally, the point will 
appear to whirl round in the opposite direction as it passes from 
one circle to the other ; whereas, if the circle B touch the circle 
A internally, the motion is the same as before, except that the 
direction of the whirl does not change. 

As a particular case let the radius of one circle become infinite, 
and we pass from circular into rectilinear motion, and conversely. 
In this case the point describing the circle goes off in a tangent, 
or the point travels along a tangent into its circular path. 

In like manner, if two circles be connected by a pair of com- 
mon tangents, as in the sketch, the point travelling round one 



i6 



Elements of Mechanism. 



circle may pass off in a tangent, describe a portion of the second 
circle, and so return by a second tangent to its primary path. 




FIG. 15. 



It is apparent that the direction of the whirl in B will be the 
same as that in A when the tangents do not cross, and will be in 
the opposite direction when they do cross. 

ART. 13. When a point passes from the circle A to the circle 
B in any of the previous cases, and is moving uniformly, the angles 
described in any given time are inversely as the radii of the circles. 
Let a point moving uniformly round the circumference of the 
circle whose centre is A pass on at 
D into the circumference of the 
circle whose centre is B without 
change of velocity. 

Let PD, DQ be arcs described 
in equal times, and let AD= r , 
DB=, PAD=0, DBQ=f 
Then DP=0.9, DQ=<ty. 
But DP=DQ, because the velo- 
city of the moving point remains unaltered, 

.-.,*=*, .ad -*f9, 

9 a AP 

which proves the proposition. '* 

ART. 14. Hitherto the motion of a point has been regarded 
under three aspects, viz. 

I. The motion in a straight line. 
II. The motion in a circle or in any other plane curve. 

III. The motion in a curve of any kind not lying in one plane. 

The like motions, when applied to material bodies, give rise to 
elementary combinations or pairs, as they are termed, which are 
repeated in an endless variety of forms in complete machines. 
At present we confine our attention to cases I and II. 





Pairs of Elements. 1 7 

Case I. A body may be moved in such a manner that every 
point of it is constrained to describe a straight line, and that all 
such straight lines are parallel to each other. 

To effect this in the most simple manner we require a pair of 
elements, viz. a sliding bar and an enveloping guide. Let the 
sliding bar be rectangular in section, as 
AB in the diagram, and have four plane 
faces. Also let it be enclosed in an 
enveloping block, with like internal 
plane surfaces, such as CD. The bar 
and the block may be conveniently 

made of cast iron, the latter being formed in two parts, and the 
surfaces being prepared by the operation of scraping. It is clear 
that the prism can only move in the direction of its axis, and that 
the motion of AB is one of simple translation, each point in the 
solid describing a straight line in the same direction. Also it is 
essential to have a pair of elements in order to provide that the 
motion shall take place in one direction only. 

Case II. The body may be constrained to move in such a 
manner that any point in it describes a. plane curve. Here, again, 
we may begin with a Dimple case, and suppose that the body is a 
pulley or wheel having a fixed axis on which it turns, and that any 
point in the rotating body describes a circle. 

Note. The term axis denotes the central line of a cylinder, 
and is a mathematical phrase : an engineer distinguishes a heavy 
cylindrical piece of metal as shafting, or a shaft, and designates 
smaller cylindrical bars as spindles ; a wheelwright speaks of the 
axle of a wheel, and a watchmaker calls the same thing an arbor. 

To obtain one motion of the second case a pair of elements io 
again necessary ; there is (i) the axle of the pulley or wheel, (2) 
the bearing or block in which the axle rotates. 

The axle of a wheel is commonly a cylindrical bar running 
completely through the wheel, and supported at both ends on 
cylindrical hollow bearings which fit the shaft and envelope it 
closely, any motion endways being prevented by collars or shoulders 
upon the shaft. 

The drawing shows the poppet head of a lathe ; the speed 
pulley which takes the driving cord is shown in section, and is 
C 




1 8 Elements of Mechanism. 

grooved in steps, the spindle or mandrel which supports the 

pulley being coned at one end 
in a steel point, and resting 
on a parallel bearing with a 
shoulder at the other extremity. 
A small hole is truly drilled in 
the direction of the axis of the 
mandrel, to receive the point 
of the cone, and to preserve 
the truth of the geometrical 
axis as the point wears.. The 

two supports form essentially one bearing, for it is apparent that 
a single cylindrical bearing, if sufficiently lengthened, would answer 
the purpose so far as rotation is concerned. 

ART. 15. In order to apply measurement to movements ot rota- 
tion it is necessary to introduce a new method of estimating velocity. 
Thus let a plane circular disc rotate upon 
an axis passing through its centre C and 
perpendicular to its plane, and let the 
radius CA come into the position CP at 
the end of a given interval of time. 

Taking CA as a fixed line of reference, 
the rate of change of the angle PCA is 
called the angular velocity of the disc. 

Angular velocity, when uniform, is 
measured by the angle described in a unit of time. The unit of 
time is always one second, unless the contrary be expressed. 

Thus, if PCA be the angle described by CP in one second, 
the angular velocity of CP is expressed by the angle PCA. 

This angle is not estimated in degrees, but in circular measure 
that is, by the ratio of AP to AC and it has been customary 
to designate it by the Greek letter w (omega). 

A P 

Hence, - 

Let r be the radius of the circle APC, v the linear velocity of 
the point P, 

thenz> = AP, w =^=*L. 
AC 7* 




Angular Velocity. 19 

which is the equation connecting the uniform linear velocity of the 
point P with the angular velocity of the disc or rotating body. 
It follows that when a body is rotating uniformly, the linear velo- 
city of any point of it increases directly as the distance from the 
axis of rotation. 

Ex. i. A wheel 6 feet in diameter turns uniformly on its centre 
20 times in a minute : what is the linear velocity of a point in its 
circumference ? 

We shall now apply the notation of foot-seconds, which are 
written f.s. for brevity. 

Here the angular velocity = "" = ^-, 
60 3 

and the linear velocity of a point at a distance of 3 feet from the 
centre of rotation 

= x 3 foot-seconds, 

= 27T f.S. 

Ex. 2. How far from the centre will a point lie which is 
moving at the rate of one mile per hour ? 

The linear velocity of this point = LL^. 3__ f. s . 

60x60 

The angular velocity of this point 

o 

/. required distance = I76ox g x -Lfeet, 

60 X 60 2JT 

= == J- feet nearly, 
ion- 10 

ART. 1 6. Again, if 6 be the angle described by CP in / 
seconds, we have tat, or w = -. 

If the angular velocity be variable, it may be proved by reason- 
ing precisely similar to that adopted in Art. 3, that the angular 
velocity w is given by the equation 

d\ 
" = dt ] 

that is to say, - J. represents the angular velocity of a rotating 
at 

body when the motion is variable. 
c2 



20 



Elements of Mechanism. 



ART. 17. Of two moving pieces that which transmits motion 
is termed the driver, and that which receives it is the folloiver. 

Conceive that the driver and follower have each a simple 
motion, either of translation or rotation ; then the ratio of their 
comparative velocities is called the velocity ratio between them. 

ART. 1 8. It will now become necessary to consider the- manner 
in which a motion of rotation of a solid body may be transferred 
from one axis to another, and it is apparent that the most simple 
case occurs when a circular disc or plate moves another in its own 
plane by rolling contact. 

In such a case the uniform motion of the axis A conveys a 
perfectly even and uniform 
motion to the other axis B. 

If A and B were circular 
plates with flat edges, and very 
accurately adjusted, it would 
be quite possible for A to 
move B by friction alone, the 
two plates rolling smoothly and 
evenly upon each other with- 
out any slipping of the surfaces in contact ; but we could not 
expect A to overcome any great resistance to motion in B ; or, in 
other words, we could not in practice convey any considerable 
amount of force by the action of one disc upon the other. 

The transmission of energy being an essential condition in 
machinery, the discs A and B are provided with teeth, as in the 

annexed figure, and the 
machariv/endeavours so 

A J l fVJ ^ " 1*7^ to form and shape the 

teeth that the motion shall 
be exactly the same as if 
one circle rolled upon 
another. 

Herein consists the 
perfection of wheelwork : 
a perfectly uniform motion 
of the axis A is to be con- 
veyed by means of teeth to the axis B ; and the motion of B, when 






Spur Wheels. 21 

tested with microscopic accuracy, is to be no less even and uniform 
than that of A. 

Since, then, it appears that the motions of A and B are exactly 
the same as those of two circles 
rolling upon each other, such imagi- 
nary circles may always be conceived 
to exist, and are called the pitch 
circles of the wheels in question. 
They are represented by the dotted 
lines in fig. 21. 

The pitch circle of a toothed 
wheel is an important element, and 
determines its value in transmitting 
motion. 

Suppose that two axes at a distance of 10 inches are to be 
connected by wheelwork, and are required to revolve with veloci- 
ties in the proportion of 3 to 2. Two circles, centred upon the 
respective axes, and having radii 4 and 6 inches, would, by rolling 
contact, move with the desired relative velocity, and would, in 
fact, be the pitch circles of the wheels when made. So that what- 
ever may be the forms of the teeth upon the wheels to be con- 
structed, the pitch circles are determined beforehand, and must 
have the proportion already stated. 

It appears also that when the number of teeth upon a wheel 
is indefinitely increased, the wheel itself degenerates into the 
pitch circle. 

So much of the tooth as lies within the pitch circle is called 
its root QI flank, and the portion beyond the pitch circle is called 
the point or addendum. 

The pitch of a tooth is the space ac upon the pitch circle cut 
off by the corresponding edges of two consecutive teeth. 

Spur wheels are represented in fig. 20, and are those in which 
the teeth project radially along the circumference. 

In a. face wheel, cogs or pins, acting as teeth, are fastened per- 
pendicularly to the plane of the wheel ; in a crown wheel the 
teeth are cut upon the edge of a circular band ; and annular 
wheels have the teeth formed upon the inside of an annulus or 
ring, instead of upon the outer circumference, 



22 



Elements of Mechanism. 



A straight bar provided with teeth is called a rack, and a wheel 
with a small number of teeth is termed opinion. 

Gearing and gear are the words used to indicate the combina- 
tion of any number of parts in a machine which are employed for 
a common object. 

Toothed wheels are said to be in gear when they are capable 
of moving each other, and out of gear when they are shifted into 
a position where the teeth cease to act. 

ART. 19. The spur wheels, before described, are suited to con- 
vey motion only between parallel axes ;_it often happens, however, 
that the axes concerned in any movement are not parallel, and as 



FIG. 22 




a consequence they may, or may not, meet in a point. If the 
axes do not intersect we proceed by successive steps, and con- 
tinually introduce intermediate intersecting axes, and thus we are 
led to the use of inclined wheels whose axes meet each other, and 
which are known as bevel wheels. 

It is easily proved in geometry that two right cones which 
have a common vertex will roll upon each other, and the same 
would be true of the frusta of two cones such as LM and NR, 
which are represented as having a common vertex in the point O. 

The rolling of the cones will allow us to consider any pair of 
circles in contact and perpendicular to the respective axes as the 
pitch circles of the frusta, and teeth may accordingly be shaped 
upon them so as to produce the same even motion as that which 
exists in the case of spur wheels. 

This fact about the rolling of two cones becomes very clear 



Rolling Circles. 23 

when enquired into, and it is evident that if one of the cones be 
flattened out into a plane table, by increasing its vertical angle up 
to 1 80, the property of rolling will not be interfered with. But 
in that case the common vertex will be a fixed point in the table, 
and, accordingly, if we roll a cone upon a table, the vertex ought 
not to move in the least degree as the cone runs round. 

It is quite easy to test the matter in this way, and if the table 
be smooth and level the apex will remain perfectly stationary, 
although the com- itself is free to run in any direction. 

The principle under discussion is sometimes applied in the 
construction of machinery ; there is a large circular saw in the 
arsenal at Woolwich which is driven by the rolling contact of the 
frusta of two cones, and upon examination it will be found that 
the directions of the axes of the two frusta meet exactly in the 
centre of the revolving circular saw. 

Equal bevel wheels whose axes are at right angles are termed 
mitre wheels. 

ART. 20. Prop. When two circles roll together, their uniform 
angular velocities are inversely as the radii of the circles. 

This proposition is exactly analogous to that which obtains 
when a point describing the circumference of one circle passes off 
into another circle of different diameter, and the proof is the same. 

Let the circles centred at A and B move by rolling through 
the corresponding angles PAD 
and QBD. 

Let AD=rt "I PAD=9 1 



then PD=flfl, QD=A/>, 
but PD=QD 




a 



But the angular velocities of the circles, being uniform, are as 
the angles described by each of them in the same given time, 
. angular vel. of A__BD 
angular vel. of B a AD ' 
which proves the proposition. 



2 4 



Elements of Mechanism. 



ART. 2t. Two simple questions relating to the transfer of 
motion by wheelwork remain to be determined. 

i. Let two axes be parallel, and let m be the velocity ratio to 

be communicated between them. 

If a be the distance between the axes, and r, r 1 be the radii 
of the two pitch circles A and B, 

the condition of rolling gives us = ;? = ve ' 

r 1 m vel. of A 



Also r+r 1 



n 
and r 1 = 



.. 

m + n 

whence r and i' are known in terms of m, n, and a. 

2. Let the axes meet in a point, and let it be required to con- 
struct two cones which shall communicate the same velocity ratio 
by rolling contact. 

We now refer to fig. 24, and assume 
that DN, DM are the radii of the 
bases of the cones LCD, HCD, whose 
angular velocities are as the numbers m 
and n respectively. 

Let MCN=a, NCD=0, 
then DN^ CD sin 0, 

DM=CDsin (a-fl)-, 
PM_sin(a-Q) 
' DN sin 

But - , since the inverse ratio of the radii of the bases 

DN n 
of the cones is the velocity ratio between the axes, 

) =sin cot 0-cos , 




n sin tf 

.. 
whence tan 6 



n sin a 

COS c 

If <p be an angle such that n cos am cos </>, (m>n), 

n sin o sin a 

we have tan 6 



Pulleys and Belts. 25 

whence is expressed in a form adapted for logarithmic compu- 
tation. 

Cor. If (=QO, we have tan 0= . 
m 

ART. 22. Belts or straps, otherwise called bands, are much 
used in machinery, in order to communicate motion 
between two axes at a distance from each other. 
In this case an endless band is stretched over the 
circumference of a disc or pulley upon each axis, 
and the motion is the same as if the discs rolled 
directly upon each other. The usual form of the ^ 
pulley is shown in fig. 25. fes ^ 

It is a common practice to convey steam power 
by means of shafting a-nd wheelwork to the various 
floors of a mill, and then to distribute it to the 
separate machines by the aid of straps or belts. 

These straps adhere by friction to the surfaces of pulleys, and 
work with a smooth and noiseless action ; but they are subject to 
two principal objections, which may or may not be counterbalanced 
by their other advantages. The friction of the axes upon their 
bearings is increased by the double pull of the strap, arising from 
its tension, and there is a liability to some change in the exact- 
ness of the transmission of motion by the possible stretching or 
slipping of the band. 

The drawing shows the method of communicating motion 
from one axis to another at a distance. The diameters of the 
large and small pulleys A and B are respectively as 3 to i, and 




the result is that when A makes 40 revolutions B makes 120 revo- 
lutions. The velocity ratio is precisely the same as if A moved B 
by rolling contact. 



26 Elements of Mechanism. 

The strap may be open or crossed. In the former case A and B 
rotate in the same direction, and in the latter case they rotate in 
opposite directions, as indicated by the arrows. 

ART. 23. The term band is applied either to a flat strap or a 
round cord indifferently. The best material for round bands, such 
as are used in light machinery, is no doubt catgut, and then the 
band is fitted with a hook and eye to make it continuous. It 
must work in a pulley with a grooved rim, or it would slip off, 
and this groove prevents our shifting it easily from one pulley to 
another. The power of readily shifting a driving band is often 
an indispensable condition, and can be obtained at once by the 
use of a flat belt, which will hold on to its pulley with perfect 
security if we only take care to make the rim slightly convex, as 
shown in fig. 25, instead of being concave. No groove is neces- 
sary, or indeed admissible ; and, upon entering a workshop where 
steam power is employed, we see each machine driven by a flat 
belt riding upon one of a set of two or more pulleys with perfectly 
smooth edges. 

The belt has no tendency to slip off, and it is shifted with 
the greatest ease from one pulley to another when pressed a little 
upon the advancing side by a fork suitably placed. 

The reason for making the rim of a pulley slightly convex will 
be apparent if we examine the case of a tight belt running upon a 
revolving conical pulley. The belt 
embraces the cone, and tends to lie flat 
upon the slant surface, thus becoming 
bent into the form AB, the portion B 
being somewhat nearer to the base of 
the cone than the portion A. 

The cone, during its revolution, 
exerts an effort to carry B onward in a 
circle parallel to its base, and the con- 
sequence is that the belt tends to remain upon the slant surface 
of the cone, and to rise higher rather than to slip off. 

In like manner, if a second cone of equal size were fastened 
to the one shown in the drawing, the bases of the two cones being 
joined together, the belt would, if its length were properly ad- 
justed, work its way up to the part where the bases met, and 




Pulleys. 



27 



would ride securely upon the angular portion formed by their 
junction ; but this is the same case as that of a slightly convex 
pulley, for it is evident that a little rounding off of the angle at 
the junction of the bases would convert this portion of the double 
cone into a convex pulley. Thus the action becomes perfectly 
intelligible. 

ART, 24. The fast and loose pulleys are an adjunct of the 
driving belt. They consist of two pulleys placed side by side, as 
in fig. 28, whereof one, A, is keyed to the 
shaft, CD, to which motion is to be con- 
veyed, and the other, B, rides loose upon 
it. When the strap is shifted from the loose E 
to the fastened pulley the shaft will begin to 
rotate, otherwise it remains at rest, the loose 
pulley alone turning round. 

The shaft EF is the driver, and carries 
one broad pulley keyed upon it. 

The band is shifted by a fork, which, as 
before stated, is made to press laterally upon c 
its advancing side. 

The advancing portion of the band must 
always lie in the plane of the pulley round 
which it is wrapped, but the retreating portion may be pulled on 
one side without causing the band to leave the pulley. This rule 
applies whether the band is round or flat. 

ART. 25. It is by observing this condition that a band may 
be used to communicate motion between two axes which are not 
parallel, and which do not meet in a point. 

Problems such as these are interesting, as presenting difficul- 
ties to be overcome by a knowledge of principles. 

Suppose that we are required to arrange that a band working 
over a pulley upon one given axis shall drive another pulley upon 
an axis at right angles to the first. 

Here we intend that the pulleys should be placed one above 
the other as in the sketch. As the band goes round we have to 
provide that its advancing portion shall always lie in the plane of 
the pulley upon which it works. The easiest way of proceeding 
is to draw a straight line, AB, upon paper, and to place circles 




AB 



28 



Elements of Mechanism. 



EF and DC representing the pulleys in contact with AB upon each 
side of it. 

Draw now the lines ED, CF, to represent bands passing round 
the circles, and however you may bend the two planes containing 





the pulleys by folding the paper about AB as an edge, it is clear 
that the advancing portion of the strap will continue to lie in the 
plane of its pulley so long as the motion occurs in the direction 
of the arrows. 

Reverse the motion, and the strap will leave the pulleys at 
once. 

ART. 26. It may be useful here to enquire how the necessary 
size or strength of the strap is ascertained when energy is trans- 
mitted, and we take the following example : 

Suppose that a force capable of doing work which is techni- 
cally estimated at 5 horse-power is to be carried on by a strap 
moving with a velocity of 600 feet per minute over a suitable 
pulley. The work done by 5 horses is 5 x 33000 foot pounds 
per minute, and the work done by the strap must be the same. 

Let P be the pull upon the strap in pounds, then P x 600 is 
the work done by the strap in one minute, 



Telodynamic Transmission of Power. 29 

.'. P x 600 = 5 x 33000, 



If the velocity of a point in the strap had been reduced to 300 
feet per minute, P would have been 550 Ibs. ; if it had been in- 
creased to 3,000 feet per minute, P would have been 55 Ibs. ; and 
thus we recognise the well-known mechanical principle, that the 
slower the movement by which any given amount of energy is 
transmitted, the greater must be the strength with which the 
moving parts are constructed. 

In carrying out this principle successful attempts have recently 
been made to transmit the driving power of turbines or water- 
wheels to considerable distances, by means of a slender wire rope 
moving at a high velocity, and the method is called the telodynamic 
transmission of power. 

The first experiment was made in 1850 by Mr. C. F. Him, at 
Loyelbach, near Colmar, Alsace. A band of steel 172 yards long, 
jj 1 ^ inch thick, and 2 inches broad was slung as an endless band 
over two pulleys, each 6^ feet in diameter, which were placed at 
a distance of 84 yards, and made 120 revolutions per minute, 
giving a speed of 28 miles per hour in the band. 

There were practical objections to the use of a flat band ; 
nevertheless the plan was successfully adopted for a year and a. 
half, and transmitted 12 horse-power to 100 looms. 

Since that time the flat rope has been replaced by a round rope 
made of steel wire. 

It may be interesting to refer to some operations carried on at 
Schaff hausen, on the Upper Rhine. Here the water-power is taken 
from three ordinary vertical-flow turbines, each 9^ feet in dia- 
meter, and driven by a fall of water varying from 12 to 16 feet. 
Each turbine makes about 48 revolutions per minute, and the 
whole can develope collectively about 750 horse-power. 

The wire rope is f inch in diameter, made of the best Swedish 
iron, and having 72 wires in the rope. It starts by running upon 
pulleys driven by the turbines, and these pulleys are each 15 feet 
in diameter, and make zoo revolutions per minute, giving a linear 
velocity to the rope of about 53 miles per hour. No less than 17 
factories in different positions have been supplied with motive 



30 Elements of Mechanism. 

power from one set of turbines, and it is stated that the total 
length of transmission is 3,300 feet. 

For a more homely illustration we may mention the locomotive 
workshops of the London and North-Western Railway at Crev/e, 
where a cotton rope |ths of an inch in diameter, and weighing 
about \\ oz. per foot, has been carried along the length of a 
workshop with a velocity of 5,000 feet per minute, and so em- 
ployed for actuating a traversing crane which is adapted for lifting 
a weight of 25 tons. The velocity of 5,000 feet per minute would 
be reduced, by suitable mechanism, to that of i foot 7^ inches 
per minute, and the requisite work would be done by subjecting 
the whole cord to no greater strain than that of 109 Ibs. 

ART. 27. Guide pulleys are sometimes used, and they are 
constructed as follows : 

Conceive that a band moving in the direction of AB is to be 
diverted into another direction, CD. There are two cases to be 
considered. 

i. Let AB and CD meet in E. At the angle E place a small 
guide pulley whose plane is coincident with the plane AED. This 
pulley obeys the required condition, and will answer its purpose. 

FIG 31. FIG 32. 





2. If AB and CD do not meet, or do not meet within a 
reasonable distance : 

Draw any straight line, EF, cutting both AB and CD. In the 
plane AEF, and at the angle E, place a guide pulley, E, and do 
the same thing at F by fixing a guide pulley in the plane EFD, and 
thus the strap will be carried on. 

One advantage in the use of guide pulleys will be found in the 
fact that they enable us to overcome the inconvenience of not 
being able to reverse the motion when the planes of the pulleys 
are inclined to each other. 



Guide Pulleys. 




Thus, conceive that two pulleys work in the planes ZAx, ZAy 
inclined to each other at an angle xAy. In the line of the inter- 
section of the planes, viz., AZ, take 
any two convenient points, H and 
B, place one guide pulley at H in 
the plane CHD, and another at B 
in the plane EBF, then the band 
CHDFBE will run round the two 
main pulleys securely in either 
direction. 

This is evident, as we have done 
nothing to infringe the necessary 
condition, each advancing and re- 
treating portion of the band will, 
in both cases, be found in the plane #/ 
of the pulley upon which it rides. 

Instead of bands we may employ chains to communicate 
motion from one axis to another, and there is one instance where 
a chain is always so used, viz., in the transfer of the pull of the 
spring from the barrel to the fusee of a watch. Here the form of 
chain, is the type of most others of the heaviest construction, con- 
sisting of one flat plate or link riveted to two others, which are 
placed one above and the other below it, and thus the chain con- 
sists of one and two plate-links alternately. When a chain of this 
sort is used to transmit great force, it is called a gearing chain, and 
the open spaces formed by the two parallel links engage with pro- 
jections on the wheel or disc over which it runs, rendering it 
impossible for the chain to slip. 

One practical objection to the use of chains, where great 
accuracy is required, consists in the fact that the links are liable 
to stretch, and that the pitch or spacing may lose its exactness, 
the result being to cause jar and vibration in the working. 

ART. 28. Instead of confining the motion of a body to simple 
translation in a straight line, or to simple rotation about a fixed 
axis, we shall now suppose that the body moves by sliding along 
a plane, in such a manner that any straight line in it, as AB, 
passes into the position A'B', which lies in the same plane with 
AB, but is not parallel to it 




32 Elements of Meclianism. 

In such a case AB has a motion both of translation and rota- 
tion. Its centre describes some line straight or curved, which 
indicates a motion of translation, and at the same time the line 
itself changes the direction in which it points, or is subject also 
to a motion of rotation. 

We shall now prove that the motion of AB may be repre- 
sented by supposing it to be rota- 
FlG - 34- ting at each instant about some 

point O, which point is continu- 
ally changing its position, and is 
therefore called the instantaneous 
centre of motion. 

The curve which the instanta- 
neous centre describes is called 
the centrode, from two Greek 
words signifying ' the path of the 
centre.' 

It will be understood that if the actual motion existing at the 
time considered were to be permanent, it would be a motion of 
rotation about an axis through O, which is therefore called the 
instantaneous axis. 

In order to find the position of O, let the motion from AB 
to A'B' be infinitesimal, and join AA , BB . Bisect AA' in E, and 
draw EO perpendicular to it ; bisect also BB' in F, and draw FO 
perpendicular to it. O will mark the instantaneous axis concerned 
in the motion from AB to A'B'. 
Join OA, OA', OB, and OB'. 
Then since OA=OA', OB=OB', and AB=A B' 

/. angle AOB = angle A'OB'. 
Take away the common angle A'OB and we have 

angle AOA' = angle BOB'. 

Therefore while A is rotating about O into the position A', the 
point B is also rotating about O into the position B', or O gives 
the position of the instantaneous axis. 

ART. 29. In some simple cases, O is a fixed point, but it may 
still have the property of an instantaneous axis. This happens 
when a body is fixed to one end of a rotating arm, the centre of 
motion of the arm lying away from the body. 



Instantaneous Axis. 33 

For example, let an arm placed horizontally be made to rotate 
about a vertical axis through one end, and let a wheel B be 
locked to the other end of the arm, in such a manner that its 
plane is horizontal. 

As the arm goes round, a person inspecting the apparatus from 
a little distance will see the wheel B turning on its axis, and if he 
watches a mark upon the rim, he can entertain no doubt about 
this fact. 

The truth is that although the wheel B does not move rela- 
tively to the arm, it is, nevertheless, the subject of two distinct 
motions, whereof one consists in a rotation about an axis through 
its centre, and the other is a motion of translation, whereby the 
centre of the wheel describes a circle whose radius is the distance 
between the centre of B and the axis of rotation of the arm. 

This is an example of the resolution of a compound move- 
ment into its simple elements, and the instantaneous axis remains 
permanently in the axis about which the arm rotates. 

If the wheel B were looked at from the centre about which 
the arm revolves, no motion of rotation could be recognised. B 
would appear to have a motion of translation which carried it 
round in a circle. 

The very same thing happens in the case of the moon. As- 
tronomers tell us that only about one-half of the face of the moon 
has ever been seen by those upon the earth's surface, and they 
explain the fact by saying that the moon turns once upon its axis 
during the period of a single revolution in its orbit round the 
earth ; or, in other words, that it moves as if it were fixed to a 
rigid bar stretching from the earth to the moon. 

ART. 30. Case III. If the motion of rotation of a body about 
an axis be combined with a motion of translation of the axis itself 
in the line of its direction, any point in the body will describe a 
curve which cannot lie in one plane. 

Such a movement may be obtained from a single pair of 
elements, as in the example of a nut on a screwed bolt. 

After toothed wheels, the screw plays the most important part 

in mechanical appliances, and indeed it is difficult to over-estimate 

its value or utility.' The screw bolt and nut are used to unite the 

various parts of machinery in close and firm contact, and are 

D 



34 



Elements of Mechanism. 



FIG. 35. 






peculiarly fitted for that purpose ; then, again, the screw is em- 
ployed in the slide rest and in the planing machine to give a 
smooth longitudinal motion, the same purpose for which it aids 
the astronomer in measuring the last minute intervals which are 
recognisable in the telescope. In the screw press we rely upon it 
to transmit force, we use it in screw piles to obtain a firm founda- 
tion for piers or lighthouses, and as a propeller for ships it has 
given a new element of strength and power to our navy. 

The definitions relating to the screw are the following : 
If a horizontal line AP, which always passes through a fixed 
vertical line, be made to re- 
volve uniformly in one direc- 
tion, and at the same time to 
ascend or descend with a uni- 
form velocity, it will trace 
out a screw surface APRB, in 
the manner indicated in the 
sketch. 

The points of intersection 
of this generating line with 
any circular cylinder whose 

axis coincides with AB, will form a screw thread, PR, upon the 
surface of the cylinder. 

The//'/^ of a screw is the space along AB, through which the 
generating line moves in completing one entire revolution. 

Also AB is called the length of the screw surface APRB, and 
the angle PRQ represents the angle of the screw. 

In the diagram, AP is shown as describing a right-handed 
screw , if it revolved in the opposite direction during its descent, 
it would describe a left-handed screw. 

ART. 31. The screw thread used in machinery is a project- 
ing rim of a certain definite form, running round the cylinder, and 
obeying the same geometrical law as the ideal thread which we 
have just described. 

In practice the pitch of a screw bolt is usually estimated by 
observing the number of ridges which occur in an inch of its 
length ; thus we speak of a screw of one-eighth of an inch pitch as 
being a screw with eight threads to the inch. 




The Thread of a Screw. 



35 



FIG. 36. 



If a single thread were wound evenly round a cylinder, and 
the path of a thread marked out, we should have a single-threaded 
screw ; whereas, if two parallel threads were wound on side by 
side, we should obtain a double-threaded screw. 

The object of increasing the number of threads is to fill up the 
space which would be unoccupied if a fine thread of rapid pitch 
were traced upon a bolt, and thus to give the bolt greater strength 
in resisting any strain which tends to strip away the thread. In- 
creasing the number of threads makes no difference in the pitch of 
the screw, which is dependent on any one continuous thread of 
the combination. 

The ordinary screw-propeller is a double-bladed screw, and 
has sometimes three or even four blades, which correspond to the 
multiple threads here spoken of. 

ART. 32. The two principal forms of screw-thread used by 
engineers are the square and the V thread ; they are given in the 
sketch, and in applying them we should 
understand that there are three essential 
characters belonging to a screw-thread, 
viz., \\spitch, depth, and form; and three 
principal conditions required in a screw 
when completed, viz., power, strength, and 
durability. 

It is easy to see that no one can de- 
clare exactly what power, strength, or du- 
rability is given by a screw-thread of a 
certain pitch, depth, or form, when traced out upon a given cy- 
linder. The problem is indeterminate, and must remain so ; we 
cannot lay down any rule for determining the diameter of a screw 
bolt required for any given purpose, nor can we say what should 
be the precise form of thread. 

It is the province of practical men to determine any such 
questions when they arise, being guided in their judgment by 
experience and by certain general considerations which we propose 
now to examine. 

i. The power of a screwed bolt depends upon the pitch and 
form of the thread. 

If the screw-thread were an ideal line running round a cylin- 




36 Elements of Mechanism. 

der, the power would depend solely on the pitch, according to the 
relation given in all books on mechanics, viz. : 
weight x pitch = power x I Circumference of the circle described 
I by the end of the lever-handle. 

If the thread were square we should substitute for the ideal 
line a small strip of surface, being a portion of the screw surface 
shown in fig. 35, which would present a reaction P to the weight 
or pressure everywhere identical in direction with that which 
occurs in the case of the ideal thread. Hence, if there were no 
friction, we should lose nothing by the use of a square thread in 
the place of a line. 

A square-threaded screw is, therefore, the most powerful of all, 
and is employed commonly in screw presses. 

But if the thread were angular, the reaction Q which supports 
the weight or pressure would suffer a second deflection from the 
direction of the axis of the cylinder over and above that due to 
the pitch, by reason of the dipping of the surface of the angular 
thread, and we should be throwing away part of the force at our 
disposal in a useless tendency to burst the nut in which the screw 
works. 

In this sense, the square thread is more powerful than an 
angular or V thread of the same pitch. 

2. The strength depends on the form and 'depth. 

This statement is obvious. In a square thread half the 
material is cut away, and the resistance to any stripping of the 
thread must be less than in the case of the angular ridges. 

Again, if we deepen the thread we lessen the cylinder from 
which the screw would be torn if it gave way, and thus a deep 
thread weakens a bolt. 

3. Finally, the durability of a screw-thread depends chiefly 
upon its depth, that is, upon the amount of bearing surface ; and 
in the case of a screw which is in constant use, as, for example, in 
the slide-rest of a lathe, it would be well for the young mechanic 
to satisfy himself upon this point by ascertaining the amount of 
bearing surface given by the fine deep thread which is found upon 
the screw working in the slide-rest of a well-made lathe. 

Probably the finest specimen of minute workmanship in 
screw-cutting will be found in the screws provided by Mr. Simms 



The Endless Screw. 



37 



for moving the cross wires or web across the field of view of a 
micrometer microscope. 

There are 150 threads to the inch, the diameter of the bolt 
being about ^th of an inch ; the head of the screw is a graduated 
circle read off to 100 parts, and the movement of the wires pro- 
duced by turning the screw-head through the space of one 
graduation is quite apparent. 

Upon examining the thread with a microscope, we should see 
a fine angular screw, consisting of a number of comparatively 
deep-cut ridges, having the sides a little inclined and the edges 
rounded off. 

In the year 1841, Sir J. Whitworth proposed a uniform system 
of screw threads for bolts and screws used in fitting up steam 
engines and other machinery. This system has been adopted, and 
has given rise to the so-called Whitworth thread, about which it 
is only necessary here to say that tables are published giving the 
pitches for screws with angular threads on bolts of given diameter, 
and further that the angle of inclination of the sides of the thread 
is constant, being 55, with one-sixth of the depth rounded off at 
the top and bottom. 

ART. 33. A worm wheel is a wheel furnished with teeth set 
obliquely upon its rim, and so shaped as to be capable of engaging 
with the thread of a screw ; the revolu- 
tion of the endless screw or worm AB 
will then impart rotation to the wheel 
C, and the wheel will advance through 
one, two, or three teeth, upon each 
revolution of AB, according as the 
thread thereon traced is a single, 
double, or triple thread. 

This reduction of velocity causes 
the combination to be particularly 
valuable as a simple means of obtain- 
ing mechanical advantage, and, as we 
have stated, the number of threads 
upon the screw determines the number of teeth by which the 
wheel will advance during each revolution of AB. 

In the transmission of force the screw is always employed to 



FIG. 37. 




38 Elements of Mechanism. 

drive the wheel, and necessarily so, because the friction would 
prevent the possibility of driving the screw by means of the wheel, 
even if the loss of power were disregarded ; but in very light 
mechanism, where the friction is insensible, the wheel may drive 
the screw, and then the screw is frequently connected with a re- 
volving fly, and serves to regulate the rate at which a train of 
wheels terminating in the worm wheel may run round. 

ART. 34. The annexed lecture diagram, taken from Sir J. 
Anderson's collection, shows an endless screw and worm wheel as 
applied in lifting heavy weights. 

FIG. 38. 




The machine is called a lifting jack, and will exert considerable 
power through a space of a few inches. Sometimes an apparatus 
of this kind consists only of a screw enclosed in rigid casing, and 
rotated by a long handle, but the drawing shows a piece of me- 
chanism which is rendered more powerful by the introduction of 
a second screw and worm wheel between the lever handle and 
the weight raised. 

It will be seen that the casing encloses a vertical square- 
threaded lifting screw, having a head and claw marked SS. Upon 
the screw is fitted a strong massive nut in the form of a worm 



The Lifting Jack. 39 

wheel, one half of which is shown in either view of the apparatus, 
the remaining half being cut away in order that the disposition of 
other working parts may be better understood. 

The worm consists of an endless screw on a spindle terminat- 
ing at C, and rotated by a lever handle HH. It is apparent that 
the lever handle and worm drive the worm wheel, and further that 
the rotation of the worm wheel or nut imparts a longitudinal 
motion to the lifting screw. 

In order that the apparatus may work, provision is made that 
the lifting screw shall not rotate, the nut in which it works is fixed 
in position in the casing, and can freely turn without shifting, 
the result being that S rises or falls slowly as the handle rotates. 

An example, set out upon the diagram, and solved upon the 
principle of work, will give a better insight into the mechanical 
construction. The friction of the working parts is neglected in 
order to obtain a simple numerical result. 

Let the pitch of the lifting screw be 1-25 inches, 

let the worm wheel have 16 teeth, and 

let the circumference described by H be 87-5 inches. 

Then motion of handle after 16 tums=87'5 x 16 inches, 

= 1400 inches, 

motion of screw at same time= i '25 inches, 
/.motion of H is to motion of S as 1400 to 1^25, 

as 1 1 20 to i. 

Let pressure on handle=2o Ibs. 
/.weight raised by screw=2ox 1120 Ibs. 
= 22400 Ibs. 
= 10 tons. 

ART. 35. In concluding this chapter we may mention that 
the doctrine of resolved motion enables us to deduce the relation 
between two balancing forces when acting on the straight lever. 
This relation follows as a direct consequence of the principle of 
work. 

Let ACB be a straight lever whose fulcxum is C, and let the 
forces P and W acting perpendicularly to the lever at the points 
A and B balance each other. 

Let the lever be now tilted into the position aCl>, and draw 
am, bn, perpendiculars on ACB. Then the resolved motion am 



4 o 



Elements of Mechanism. 



represents the displacement of A in direction of P, while nb re- 
presents the displacement of B in direction of W. 

FIG. 39. But the forces P and W 

2 balance, and no work is done, 
therefore 



tf* J?-^l- 

T^ 



P*am-Wx&n=0. 

^_bn _C_CB 
01 W~^~~Crt~CA' 



FIG. 40. 



which is the well-known condition of equilibrium. 

ART. 36. Bell-crank levers serve to change the line of direc- 
tion of some small motion, and are of universal application. They 
consist simply of two arms standing out from a fixed axis so as to 
form a bent lever. 

i. Suppose it to be required to construct a bell-crank lever so 
as to change the direction of some small motion from the line BD 
into the line DA, where BD and DA 
meet in a point D. 

Draw DC, dividing the angle at D 
into two parts whose sines are in the 
ratio of the velocities of the movements 
in the given directions. 

This may be done by setting up 
perpendiculars anywhere on BD and 
DA in the required ratio, and drawing 
straight lines through their extremities parallel to BD and DA re- 
spectively. These parallel lines will intersect somewhere in DC, 
and will determine that line. 

In DC take any convenient point C, and draw CA, CB, per- 
pendicular to DA and DB respectively, then ACB will be the bell- 
crank lever required. 

This is the construction, and it can be immediately verified, 
for the arcs described by the extremities A and B, when the lever 
ACB is shifted through a small angle o, will be represented by 
AC x a and CB x a respectively, and will measure the velocities 
of the points A and B. 

TT velocity of A_ACx_AC_sin CDA 

velocity of B CBx CB~sin CDB' 




Bell-crank Levers. 



FIG. 41. 



It is evident that the movements in DA and DB are very nearly 
rectilinear, and will become more so the further we remove C from 
the point D. 

Any play which may be necessary at the joints A and B, by 
reason that the ends of the levers really describe small circular 
arcs, may be easily provided for in the actual arrangement. 

2. To change the direction from one line to another not inter- 
secting it. 

Draw PQ, a common perpendicular to the lines AD and BE ; 
through Q draw QH parallel to 
DA ; construct a bell-crank lever, 
a cb, for the movements as trans- 
ferred to the lines BQ, QH; draw 
ce parallel to PQ and equal to it, 
and further make e d parallel and 
equal to cb. 

The piece a ced will be the 
lever required; what has been 

done is this, a bell-crank lever ^ 

a cb has been formed by the rule 

given above in order to transfer the motion from BE to QH, and 
then the motion in QH has been shifted into another line DA 
parallel to itself. 




Elements of Mechanism. 



FIG. 42. 



CHAPTER II. 

ON THE CONVERSION OF CIRCULAR INTO RECIPROCATING 
MOTION. 

ART. 37. In discussing the nature of harmonic motion we 
have necessarily been led to consider the most elementary form of 
apparatus for converting the circular motion of a pin moving in 
one plane into the reciprocating motion 
of a guided bar. t 

The annexed sketch, taken from a 
model belonging to the School of Mines, 
shows an apparatus having a pin, P, 
fastened to a disc of wood, and capable 
of being rotated by a handle at the back. 
A horizontal slotted bar, EF, attached at 
right angles to a vertical guided bar, AD, 
completes the arrangement. Such an ap- 
paratus has already been referred to, and 
it is apparent that the bar rises and falls 
with a true harmonic motion as the pin, 
P, moves round uniformly in a circle. 
We pass on to analyse the conversion of circular into recipro- 
cating motion by means of the crank and connecting rod. 

A crank is merely a lever or bar movable about a centre at 
one end, and capable of being turned round by a force applied at 
the other end ; in this form it has been used from the earliest 
times as a handle to turn a wheel. When the crank is attached 
by a connecting rod to some reciprocating piece, it furnishes a 
combination which is extremely useful in machinery. 

In the next chapter we shall see that the crank and connecting 




The Crank and Connecting Rod. 43 

rod is one of the principal contrivances for converting recipro- 
cating into circular motion ; the student will understand that any 
such distinction as to the effect of the contrivance is one of classi- 
fication only, regard being had to the direction in which the 
moving force travels. The arrangement is often used under both 
aspects in one and the same machine ; as in a marine engine, 
where the piston in the steam-cylinder actuates the paddle-shaft 
by means of a crank and a connecting rod, and the motion is then 
carried on, by a crank forged upon the same shaft, to the bucket 
or piston of the air-pump. 

It was in the year 1769 that James Watt published his in- 
vention of ' A Method of Lessening the Consumption of Steam 
and Fuel in Fire-Engines,' the main feature of which was the 
condensation of the steam in a vessel distinct from the steam- 
cylinder. The steam-engine was at that time called a fire-engine, 
and was used exclusively in pumping water out of mines. The 
steam piston and the pump rods were suspended by chains from 
either end of a heavy beam centred upon an axis, the action of 
the steam caused a pull in one direction only, and the pump rods 
being raised by the agency of the steam were afterwards allowed 
to descend by their own weight. 

In this shape the steam-engine was entirely unfitted for actu- 
ating machinery, and it was not until after the impulse given by 
Watt's invention was beginning to be felt that it became apparent 
that the expansive force of steam could be made available as a 
source of power in driving the machinery of mills. 

While Watt was occupied with the great problem of the con- 
struction of double-acting engines, which eventually he fully 
solved, it happened that one James Pickard, of Birmingham, in 
the year 1780, took out a patent for a 'new invented method of 
applying steam or fire engines to the turning of wheels,' in which 
he proposed to connect the great working beam of the engine 
with a crank upon the shaft of a wheel by means of a spear or 
connecting rod, jointed at its extremities to the beam and crank 
respectively. 

It is probable that Watt had foreseen this application of the 
crank as early as the year 1778, and had intended to apply the 
combination as a means of carrying on the power from the end of 



44 



Elements of Mechanism. 



the working beam to the fly-wheel. Being forestalled, however, by 
the patentee, he did not dispute the invention, and contented 
himself with patenting certain other methods of obtaining a like 
result, among which will be found the sun and planet wheels de- 
scribed in a subsequent chapter. 

This latter invention served his purpose until the patent for 
the crank had expired, and then it was that the arrangement which 
we are now about to discuss came into general use. 

The manner of employing the crank and connecting rod in the 

FIG. 43. 




locomotive engine is shown in fig. 43. The crank CP is made a 
part of the driving wheel of the engine, the connecting rod PQ is 
attached to the end of the piston rod QR, and the end Q is 
constrained to move in a horizontal line by means of the guides 
HK, LM. 

In this engine the crank is the follower and not the driver, 
but the combination is the same whether the circular motion of 
CP causes the reciprocating motion of Q, or whether the recipro- 
cation of Q imparts a circular motion to CP. 

ART. 38. We now pass on to discuss the primary fundamental 

piece of mechanism de- 
rived from a simple tri- 
angle. 

Since a triangle is an 

immovable figure, and 

will not 'rack,' as mechanics express it, some provision must be 
made for varying the dimensions of at least one side. 



Fis. 44- 



The Crank and Connecting Rod. 45 

In our combination, the crank CP is of fixed length, as is also 
the connecting rod PQ, but the side CQ is of variable length. 

When CP performs complete revolutions, it is clear that Q 
will reciprocate in the line CQ. We shall presently see that the 
motion of Q is of an aggregate character, or that Q is the reci- 
pient of two distinct movements which are simultaneously im- 
pressed upon it. 

ART. 39. To determine the relative positions of the crank 
and connecting rod during the motion. 
FIG. 45. 




Let CP be the crank centred at C, PQ the connecting rod, and 
let the point Q be constrained to move in the straight line CED. 

Draw PN perpendicular to CD, and let CP=#, PQ=, also 
let the angle PCQ=C, and PQC=Q. 

Then CQ = CN + NQ = a cos C + cos Q. 



=4 m . sin Q sin c 

sin C b b 



2 C 



.-. CQ = a cos C + N/ 2 a a sin 2 C, 

which gives the position of Q for any value of C, that is, for any 
given position of the crank CP. 

Cor. i. LetC = o, /. CD =a+; 

C= i8o,/.CE= - a + b; 
whence DE = CD CE = 2a. 
The space DE is called the throw of the crank. 
Cor. 2. If the position of Q be estimated by its distance from 
D, we have 

DQ = CD - CQ= a + b-(a cos C+b cos Q) 
= a (i-cos C) + (i-cos Q). 



46 Elements of Mechanism. 

But we have just shown that an expression such as a (i cos C) 
represents the resolution of circular into reciprocating motion, and 
we infer that the motion of the point Q is compounded of the 
resolved parts of two circular motions, one being that due to the 
motion of P through an angle C in a circle round C, the other, 
that resulting from the motion of P in a circular arc through an 
angle Q, and produced by the swinging of the rod PQ about one 
end Q as it moves to and fro. 

Hence the connecting rod introduces an inequality, which pre- 
vents the motion of the point Q from retaining that evenness and 
regularity of change which was found in the motion of the point 
N (Art. 7). We now see by analysis that this inequality, whereby 
the motion of Q differs from that of N, is equal to b (i cos Q). 

Ex. Let CP = 10 inches, PQ 5 feet, as in the engine in 
fig. 43 ; find the position of the piston when the crank is vertical. 



-cos Q) 
= 10 + 60(1-^35) 
= 10 +'84 nearly, 

or Q is nearly six-sevenths of an inch in advance of the centre of 
its path when the crank has made a quarter of a revolution from 
the line CD. 

As the connecting rod is shortened - the inequality increases, 
and the motion becomes more unequal. 

Take a very extreme case, where PQ=CP=#, 
.-.DQ = tf(i cos C)-f a(\ cos Q). 

Let now C=6o, then Q=6oalso, because the triangle CPQ 
is now isosceles, 

/.cos C=-=cos Q 

/.DQ=20 a=a, 

or Q has moved through half its path, while CP has turned 
through an angle of only 60. 



Ratio of Velocities. 47 

When 0=90, the point Q comes to C, and there the motion 
ends, for the crank CP can now go on rotating for ever without 
tending to move Q. 

Co r. 3. If the 'connecting rod could be prolonged until it be- 
came infinite, we should have PQ always parallel to itself, or 
Q = o, and in that c;ase the travel of Q would be represented by 
the equation DQ=(i cos C). 

A crank with a connecting rod of infinite length is an imagi- 
nary creation, but we shall presently see that an equivalent motion 

e obtained in various ways. 
'ART. 40. To determine the velocity ratio of P to Q : 

V be the velocity of P, v the velocity of Q. From C 
a straight line at right angles to CQ, and let it meet QP 
produced in H (fig. 45). 

Then since PQ is a rigid rod, the resolved velocity of P along 
QP is equal to the resolved velocity of Q along QP. 

Let PCQ=C, PQC=Q, CPH=a. 

Then P may be taken to move in a tangent at P, and therefore 
its resolved velocity in PQ is V sin a, also the resolved velocity of 
Q in the same direction is v cos Q. 

Hence V sin o=z> cos Q, 
. #_sin n _sin a_CH 
' 'V~cos Q~sin H~~CP~' 

The same result may be obtained by analysis, for making 
PCQ=0, PQC- & we have 

,, a<M h d$ , a 2 sin cos <M 

V=~, ^sbflg*^ _. 



Let 
draw a 



Whence =sin0 + _ 

V N/ 



b cos <f> 
s0+ 

If COS 

sin x CQ 
PQ cos $ 

sin sino _ sin o _ CH 
cos-p sin sin H CP ' 



48 



Elements of Mechanism. 



ART. 41. The relative velocities of P and Q may be set out 
in a diagram. 

Take CP : PQ:: i I 4, which will give a figure of convenient 
dimensions, and let Q,^ be two positions of the end of the con- 
necting rod. Produce QP, qp to meet the vertical diameter FCK 

FIG. 46. 




in H and h respectively, the line BCD being horizontal. Join 
CP, C/, and measure off CS = CH, and Or = Ch. Then, as 
before, 

z>_CH_CS 

V~CP~CP" 



In like manner, 



-=--, and so for all other values, 



vel. of p C/ 

whereby the curve CS.y will indicate the comparative velocities of 
the crank pin and of the end of the connecting rod during one 
single stroke. Also, by a similar construction, the lower loop may 
be set out, preserving the proportion, 

Vel. of r:vz\. of R::C/: CR, 
and making CT = Ct. 

If PQ were kept parallel to itself throughout the motion we 
should have in effect a crank with an infinite link, and the looped 
curve would become a circle described on the radius FC as a 
diameter, as in the case of the simple harmonic motion. The in- 
equality introduced by the obliquity of the connecting rod is thus 
rendered apparent by comparison of the distorted loop with the 
true circle. 

Ex. To compare the positions of Q when CP makes a given 
angle of x with CA, CB respectively. 



The Eccentric Circle. 



49 



Let D and E be the extreme positions of Q, then 

DQ=0(i cos Q} + b(\ cos </>) 

EQ=20 DQ=0(i+cos Q} b(i cos </>). 

Also, let d be the circular measure of x, then $ can be found 
in terms of a, b, and 0. 

Take the case of a direct acting engine, where a = i foot, 
b = 6 feet, and letjc=i. Calling Q,Q' the required positions 
of the end of the connecting rod, we have cos 0= -9998477, 
cos < = -9999958, 

/.DQ = -oooi775 feet, 
EQ'=-oooi27i feet. 

ART.' 42. The eccentric circle supplies a ready method of ob~ 
taining the motion given by a crank and link, and we proceed to 
examine it with the intention of ascertaining by what expedients 
we are enabled to vary the lengths of the particular crank and link 
which exist in every form of the arrangement. 

And, first, we notice that the length of the crank is in every 
case equal to the distance between the centre of F|G 47 

motion and the centre of the eccentric circle ; 
it is, in fact, the line CP in each of the annexed 
drawings. 

Let us consider the motion shown in fig. 47, 
where a circular plate, movable about a centre 
of motion at C, imparts an oscillatory movement 
to a bar, QD, which is capable of sliding between 
guides in a vertical line, DQ, pointing towards / 
C. Since PQ remains constant as the plate re- / 




volves, it is evident that Q moves up and down 

in the line CD, just as if it were actuated by the 

crank, CP, and the connecting rod, PQ. The 

length of the connecting rod is in this case, 

therefore, equal to the radius of the rotating circle. It is obvious 

that an arrangement of this kind would be little used, by reason 

of the oblique thrust on the bar QD. 

A second form which, however, is of the greatest possible value, 
is deducible at once from that last examined. 




5O Elements of Mechanism. 

Instead of allowing the end of the bar, QD, to rest directly 
FJG g upon the circumference of the circle, suppose 

that bar to terminate in a half-hoop which fits 
the circle, as shown in the drawing ; let the rod 
point to the centre of the circle, and let one 
end, Q, be compelled to move in a line point- 
ing to C. As the circle revolves it is evident 
that we have a crank, CP, just as before ; but 
we have, in addition, a link which is now re- 
presented by PQ, and which may extend as far 
as we please beyond the limits of the circular 
plate. 

We thus obtain a combination which is 
sometimes described as a mechanical equiva- 
lent for the crank and connecting rod. 
The form usually adopted in practice is derived at once from 
the arrangement just described. A circular plate is completely 
encircled by a hoop, to which a bar is attached ; this bar always 
points to P, the centre of the plate, and its extremity drives a pin, 
Q, which is constrained to move in the line CQ. 

The plate is movable about a centre of motion at C, and we 
have already explained that PQ remains constant during each re- 
volution of the plate, or that the resulting motion impressed upon 
Q is that due to a crank, CP, and a link, PQ. 

As before, the throw of the eccentric is the same as that of the 
crank, viz., a space equal to the diameter of the circle whose 
radius is CP. 

We should remark that P, the centre of the plate, may be 
brought as near as we please to C, the centre of the shaft, and 
that the throw of the eccentric may be reduced accordingly ; but 
that we are limited in the other direction, for the shaft must be 
kept within the boundary of the plate, and the plate itself must 
not be inconveniently large, considerations which are sufficient 
to prevent our increasing CP in any great degree. 

The eccentric circle may also be regarded as a simple form of 
cam (see Art. 58), but we have examined it here on account of its 
being identical in principle with the crank and connecting rod. 
The object of the complete hoop is to drive Q in alternate 



The Eccentric. 



directions. In some cases Q is brought back by a spring, and 
then only half the hoop is required. An instance occurs in modern 
forging machines, where the motion is very small and rapid. 

ART. 43. Having thus explained the principle of construc- 
tion adopted in the eccentric, it remains 
to show the contrivance as made and 
applied in an engine. In the annexed 
diagram, which is taken from a small 
oscillating engine, the circle C repre- 
sents a section of the crank shatt, C 
being its centre. Upon the shaft are 
fitted two circular half-pulleys of cast 
iron, which are bolted together, and 
have a centre at P. Two half-hoops of 
brass, tinted in the sketch, and united 
together by bolts and double nuts at E 
and H, carry the eccentric bar, which 
actuates a pin at B connected with the 
valve lever. The engine being designed 
for a river boat, and therefore requiring 
to be reversed at pleasure, there is a 
strap, ab, to prevent the eccentric rod 
from falling away from the pin while 
the valve is being moved by hand. 
Also, in this case, the eccentric pulley 
rides loose upon the shaft within cer- 
tain limits defined by stops, and there is consequently a disc, D, 
forming a counterbalance to the weight of the pulley, which pr,e- 
vents it from falling out of position during the disengagement of 
the pin at B. 

ART. 44. It will be understood that the crank and connect 
ing rod labour under the disadvantage of entailing a division ot 
the shaft whenever it is required to place the crank anywhere ex- 
cept at one end, for the connecting rod is continually traversing 
over the centre of the shaft. 

If, therefore, a crank is wanted in some intermediate portion 
of an axle or shaft, the axle must be cranked in the manner shown 
in fig. 50, or be divided, and the two cranks or arms will be con- 




Elements of Mechanism. 



FIG. 50. 



nected by a pin. These cranks and the pin are frequently forged 
in one solid mass upon the shaft, and shaped afterwards by the 
machinery of the workshop. 

The annexed sketch is taken from a lecture diagram in Sir J. 
Anderson's collection, and repre- 
sents a small vertical engine suitable 
for driving light machinery. 

The steam cylinder is marked C, 
and H is the slide case, the piston 
rod being connected with the crank 
pin by the connecting rod PR. The 
slide valve is worked by an eccentric, 
shown at E, and the eccentric rod 
attached to the valve spindle is 
marked ED. 

The great value of the eccentric 
arises from the circumstance that it 
enables us to derive the motion, 
which would be given by a crank, 
from any part of a shaft without the 
necessity of subdividing it. This is 
particularly noticeable in the me- 
chanism here set forth, where the 
crank of small throw which is re- 
quired for moving the steam slide- 
valve is furnished by the aid of an 
eccentric keyed upon the main 
shaft. 

\\RT. 45. It is hardly necessary to explain that when a circle 
revolves about an axis perpendicular to its plane, and a little out 
of the centre, it will be enveloped in all positions by a somewhat 
larger circle, the increase of the radius being equal to the eccen- 
tricity. 

This fact has been applied to a useful purpose in the produc- 
tion of gun-stocks by machinery, as, for example, at the Small 
Arms Factory at Enfield. The practice has been to form a har- 
dened steel block with cavities, shaped so as exactly to correspond 
with the cavities which it is intended to carve out of the gun- 




The Eccentric Circle. 53 

stock. The clearing out of the recesses is effected by revolving 
drills, making some 6,000 revolutions per minute, which are 
carried over the work, and are guided on the copying principle by 
a dumb tracer, which travels over every portion of the steel block. 
The tracer and the rotating drill exactly correspond in size. For 
more delicate portions of the work smaller bits with corresponding 
tracers are employed, and thus the wood is carved out with the 
details of form which are to be found in the pattern. 

But now comes a difficulty. The tracer and pattern are of 
hardened steel, and each tracer has its corresponding bit or drill, 
which is an exact revolving counterpart thereof, so long as its 
cutting edges are not worn away. But when the drills wear by 
sharpening, and become smaller, the copy will be defective. 

In order to compensate for this source of error, the conical 
hole in the end of the running spindle into which the drills are 
fitted is made slightly eccentric. The 
drill is also eccentric on its shank to the FlG - si- 

same amount. 

It is therefore possible to bring the 
axis of the drill itself exactly in the same 
line as the axis of the running spindle, 
in which case the drill will revolve in a 
cylinder of its own size. Or the axis of 
the drill may be set on one side of the 
axis of the running spindle, in which case the drill will carve out 
a cylinder larger than itself. 

The end of the spindle and drill are graduated round the 
circumference, and thus the amount of eccentricity, and the con- 
sequent enlargement of the drill spindle, can be adjusted with the 
utmost nicety. 

In the drawing A represents the centre of the running spindle, 
B is the centre of the conical hole made therein, C is the centre 
of the drill spindle. When the drill is at work C describes a circle 
round A. But C can be shifted into any position round B as a 
centre, and therefore C can be brought either to coincide with A, 
or can be placed at any distance from A, lying between the limits 
zero and AC. Hence the result which has been stated. 

ART. 46. There is yet another method of arranging the eccen- 




.94 



Elements of Mechanism. 




trie circle, which gives us the combination of a crank with an 
infinite link. 

We here intend, as explained in a former article, that the re- 
ciprocating piece shall be driven by a crank and connecting rod in 
such a manner that the connecting rod 
shall always remain parallel to itself, a re- 
sult v hich could only happen in theory 
if the connecting rod were indefinitely 
lengthened. 

Suppose the roller at Q to be replaced 
by a cross bar QR, standing at right 
angles to DC. As the circle revolves, it 
will cause the bar to reciprocate, CP will 
remain constant, and PQ will always be 
at right angles to RQ, and will therefore 
remain parallel to CD in all positions. 

But this is the motion of a crank with 
an infinite link. 

If C were in the circumference of the circle, the motion would 
be just the same, except that now the crank would be the radius 
of the eccentric circle. We shall presently notice a useful illus- 
tration of this particular case. (Art. 48.) 

Take the following as an example of the movement under dis- 
cussion. 

In Sir J. Anderson's machine for compressing elongated rifle 
bullets, which was in use during the Crimean war, there are 
punches fixed at the two ends of a strong massive rod, to which a 
reciprocating motion in a horizontal line is imparted, and a piece of 
lead is compressed into the required form at each end alternately. 
The object of one part of the mechanism is to cause this rod 
to reciprocate, and the movement is obtained as follows. 

A small circle centred at C represents in section a shaft caused 
to rotate by the power of an engine. Upon this shaft a short 
cylindrical block is forged so as to form part of- it, and is after- 
wards accurately turned into the form of a circular cylinder, whose 
axis passes through P upon one side of the original axis. 

A rectangular brass block, RS, is bored out to fit the larger 
cylinder and slides in the rectangular frame FG, to which the 



Examples of Harmonic Motion. 



55 



cylindrical pieces AB, DE, which carry the punches, are attached. 
The whole is put together in the manner shown, and it requires 
very little effort to understand that the rotation of the eccentric 
cylinder round an axis through C will impart to RS simultaneous 



FIG. 53- 




movements in a horizontal and vertical direction, whereof one, 
viz., that in a direction perpendicular to AE, will be inoperative, 
and the other will be communicated directly to FG, and so to the 
rods carrying the punches. Thus AB and ED will be made to 
oscillate in the guides indicated in the sketch. In this case, then, 
the eccentric circle, whose centre is at P, is caused to rotate about 
the point C, and gives motion to the sides of the frame FG, just 
as it moved the bar QR in the last article, and hence the resulting 
motion is that of a crank, CP, with an infinite link. 

The contrivance of the ratchet wheel at the right hand will be 
explained in a subsequent chapter. 

ART. 47. The crank with an infinite link also appears under 
the guise of a swash-plate. 

Here a circular plate, EF (fig. 54), is set 
obliquely upon an axis, AC, and by its rotation 
causes a sliding bar, PQ, whose direction is 
parallel to AC, to oscillate continually with an 
up and down movement, the friction between 
the end of the bar and the plate being relieved 
by a small roller. 

We must now try and ascertain what is the 
law which governs this motion, and we observe that since PQ 



FIG. 54. 




5G Elements of Mechanism. 

remains always parallel to AC, the actual path of Q, as projected 
upon an imaginary plane through the lowest position of Q, and 
perpendicular to AC, will be a circle. 

If this be so, it follows that the path of Q upon the plate 
itself will not be a circle, but an oval curve, and, as a matter of 
geometry, we can prove that the line CQ will vary in length as Q 
rises or falls during the rotation of the plate, in the precise 
degree necessary for the description of the curve known as an 
ellipse. 

In fig. 55 let EQF represent the actual path of Q on the plate, 
and let the circle ERD be the projection of 
this path upon a plane perpendicular to the 
axis AC. 

Draw QM perpendicular to EF, QR 
perpendicular to the plane ERD, and RN 
perpendicular to ED, which is the diameter 
of the circle ERD. 
Join MN, and suppose the plate to rotate through an angle 
EAR = A, and thus to carry the roller at Q through a vertical 
space equal to RQ. 

Then RQ = MN = AC x 




= AC(i -cos A); 
or the motion is that of a crank AC with an infinite link. 

This is a curious result. Hitherto, in the motion of a crank 
with an infinite link, the reciprocation has always taken place in 
a plane perpendicular to the axis of rotation, but here we get the 
very same movement in a plane which contains the axis instead of 
being perpendicular to it. 

ART. 48 It is sometimes required that the reciprocating mo 
tion shall be intermittent, or have intervals of rest. 

This motion may be provided for by placing a loop at the point 
where the eccentric bar engages the pin. It is evident that the 
pin will only move when one end of the loop takes it up but in 



Intermittent Motion. 



57 



FIG. 56. 



doing this a blow is struck, which it may be well to avoid, and 
hence an intermittent motion has been obtained in a much better 
manner by a movement adapted for working the slide-valves of a 
steam-engine. 

We can readily see that if any portion of the plate in Art. 42 
be shaped in the form of a circle round C, 
such portion will have no power of moving 
the sliding bar. 

Let the pin P assume the form of a 
circular equilateral triangle, CAB, formed 
by three circular arcs, whose centres are 
in A, B, and C respectively, and let it be 
embraced by a rectangular frame attached 
to a sliding rod. 

As CAB revolves round the point C, 
the portion CB will raise the plate ; the 
point B will next come into action, and 
will raise the plate still higher ; the upper 
edge of the groove will then continue for 
a time upon the curved surface AB, which 
is a circular arc described about C as a 

centre, and here the motion will cease ; the plate will next begin 
to fall, will descend as it rose, an interval of rest will succeed, and 
thus we shall produce an intermittent movement, which may be 
analysed as follows : 

Suppose the circle described by B to be divided into six equal 
parts, at the points numbered i, 2, 3, 4, 5, 6. 

FIG. 57. 






As B moves from i to 2, the frame remains at rest ; from 2 to 



58 Elements of Mechanism. 

3 the arc CB drives the frame, the centre of motion of the eccen- 
tric circle being now a point in its circumference, and the hori- 
zontal bar is driven as it would be by a crank CB with an infinite 

FIG. 58. 





link (see Art. 46) ; from 3 to 4 the point B drives, and the motion 
is again that of a crank, CB, with an infinite link (see Art. 46) ; i.e., 
the motion from 3 to 4 is the same as that from 2 to 3, except that 
it is decreasing in velocity instead of increasing. 

From 4 to 5 there is rest, then an increase of motion from 5 to 
6, and finally a decrease to zero as B passes through the arc 6 to i 
and completes an entire revolution. 

ART. 49. Circular may be converted into reciprocating motion 
by the aid of escapements. 

An escapement consists of a wheel fitted with teeth which are 
made to act upon two distinct pieces or pallets attached to a re- 
ciprocating frame, and it is arranged that when one tooth escapes, 
or ceases to drive a pallet, the other shall commence its action. 

One of the most simple forms is the following : 

A sliding frame, AB, is furnished with two projecting pieces at 
C and D, and within it is centred a wheel possessing three teeth, 
P, Q, and R, which tends always to turn in the direction indicated 
by the arrow. 

FIG. 59. 




The upper tooth, P, is represented as pressing upon the pro- 
jection D, and driving the frame to the right hand : when the tooth 



Escapements. 



59 



P escapes, the action of Q commences upon the other side of the 
frame, and the projection C is driven to the left hand. Thus the 
rotation of the wheel causes a reciprocating movement in the 
sliding piece, AB. 

It is clear that the wheel must have i, 3, 5, or some odd num- 
ber of teeth upon its circumference. 

ART. 50. The Crown Wheel escapement was invented for 
the earliest clock of which we possess any record. 

The form of the wheel is that of a circular band, with large 
saw-shaped teeth cut upon one edge ; the vibrating axis, AB, 
carries two flat pieces of steel, a, b, called 
pallets, which project from the axis in 
directions at right angles to each other, 
and engage alternately with teeth upon 
the opposite sides of the wheel. In order 
to ensure that each pallet shall be struck 
in succession, either the wheel musbhave 
an odd number of teeth, or the axis of 
the pallets must be set a little out of the 
central line. Suppose the wheel to turn 
in the direction towards which the teeth 
incline, and let one of its teeth encoun- 
ter the pallet b and push it out of the 
way ; as soon as b escapes, a tooth on the 
opposite side meets the pallet a and 
tends to bring the axis AB back again : thus a reciprocating 
action is set up, which will be very rapid unless AB is provided 
with a heavy arm, CD, at right angles to itself. Such an arm 
possesses inertia, so that its motion cannot be suddenly checked 
and reversed, and a recoil action is set up in the wheel, which 
materially subtracts from the utility of the contrivance. For it 
will be seen that the vibration of CD cannot be made to cease 
suddenly, and that the wheel must of necessity give way and re- 
coil at the first instant of each engagement between a tooth and 
its corresponding pallet. 

The more heavily CD is loaded at a distance from the axis the 
more slowly will the escapement work, and the greater will be the 
amount of the recoil. 




6o 



Elements of Mechanism. 



Here we have an invention which has done good service to 
mankind. It was used in the first clock which was ever made, and 
dealt out time through the step-by-step movement of the wheel with 
pointed teeth. This wheel, urged on by a weight, and hampered 
always by the vibrating bar, whose pallets were perpetually getting 
into the way of its teeth, moved round with a slow, intermittent, 
and step- by-step movement, checked and advancing alternately, 
but solving for mankind, in a clumsy though tolerably accurate 
manner, the great problem of the mechanical measurement of 
time, and giving birth, by the idea suggested, to those marvellous 
pieces of mechanism which have finally resulted in the modern 
astronomical clock and the chronometer. 

Being, however, in some particulars, a defective or imperfect 
contrivance, it has gradually sunk from one level to another ; it 
has disappeared from clocks and from watches also, and is now 
seldom to be met with except in the homely contrivance of the 
kitchen-jack for roasting meat. 

ART. 51. We purpose, before going further, to examine a 
little more particularly the mechanism of an escapement, so as to 
gain some idea of the refinements of its construction when ap- 
plied to the best made clocks or watches, and we will review very 
rapidly the elementary facts which are to be found in the books 
on mechanics. 

An imaginary or simple pendulum is a conception of mathema- 
ticians, and is defined as being a single particle of matter, P, sus- 
pended by a string, DP, without weight. 

FIG. 61. This particle may swing to and fro in a 

vertical plane under the action of the pull of 
the earth, and the oscillation of the pendulum 
is the whole movement which it makes in one 
direction before it begins to return, viz., ACB 
The time of a small oscillation is the period 
of this movement, and is given by the for- 
mula : 




where / = time of a swing in seconds, 
length of the string in feet, ^=32-2 feet. 



The Pendulum. 6 1 

Make /= i, and we have the length of a pendulum oscillating 
seconds, which is a little less than a metre, being equal to 39 -14 
inches. 

The discovery of the so-called pendulum law was made by 
Galileo, who noticed that a lamp swinging by a chain in the me- 
tropolitan church at Pisa made each movement in the same time, 
although gradually oscillating through smaller arcs while coming 
to rest. 

For ordinary purposes the time of a swing may be supposed to 
depend only on the length of the pendulum, and not at all upon 
the arc through which it swings. It is practically very nearly the 
same for arcs up to 2 or 3 degrees on each side of the vertical, 
and it may be shown by calculation that the error introduced by 
assuming the law to be strictly true, in the case of a pendulum 
moving through an arc of 5 degrees from the vertical position, 
would only amount to 2rirVo tn P art f tne ^ me f a swing. 

A seconds' pendulum in a well-made clock swings through 
about 2 degrees on each side of the vertical. 

The question now arises, how is the law of the swing of this 
imaginary pendulum to be applied in the regulation of clocks, and 
wherein does a solid heavy pendulum, which we must of necessity 
employ, resemble or differ from an ideal pendulum? 

Conceive a straight uniform bar of iron to be suspended close 
to one end upon small triangular wedges of hardened steel, 
technically called knife edges, which rest upon perfectly horizontal 
agate or steel planes, and then to be set swinging. It is evident 
that each particle of the bar will endeavour to observe the pen- 
dulum law stated above, and will tend to swing in different times. 
But all the particles must swing together, and the result is that a 
sort of compromise takes place between the different tendencies, 
and the \vhole bar swings as if its material were all collected into 
one dense point at a certain distance from the knife edge, which 
is known as the centre of oscillation. Thus the solid pendulum 
swings as an ideal pendulum would do whose length was equal to 
the distance between the centres of suspension and oscillation. 

This is the theory of the rigid pendulum, which has been in- 
vestigated by Huyghens, and by others after him, and has led to 
many interesting experiments in applied mechanics. 



62 Elements of Mechanism. 

ART. 52. A method of showing roughly to a class the position 
of the centre of oscillation is the following : 

The drawing represents a wooden sword, having a pin at C 
passed through its handle and resting in two small upright sup- 
ports, in such a manner that the sword can freely turn in a vertical 
plane about the point C. 



The sword is divided into two parts at B in the proportion 
show r n, and there is a tightening screw at B, which may be set to 
make it somewhat difficult to bend the sword out of shape. The 
sword is then allowed to hang freely as a pendulum from C, and a 
ball suspended by a silk thread at C is swung by the side of it. 
When the sword and the bullet swing together the position of the 
centre of the bullet is marked at O upon the sword. 

The experiment can now be completed. An anvil, P, is placed 
under the sword as in the figure, the sword is raised, and is allowed 
to fall and strike upon P. It will bend as shown. If CP be 
greater than CO it will bend with its end upwards, whereas if CP 
be less than CO it will bend downwards, just as if CA were a thin 
wire carrying a weight, O, upon it. The whole weight of the sword 
is for the purpose of the blow collected in the point O, and if CP 
be equal to CO the sword will not bend at all. 

ART. 53. The application of the pendulum in the manner in 
which it is now employed in clockwork forms an important branch 
of mechanism, and a great advance was made by Dr. Hooke, 
the contemporary of Newton, who devised the so-called Anchor 
Escapement. 

Looking at the contrivance without regard to its connection 
with a pendulum, we find in it a wheel with pointed teeth, which 
is centred at E, and tends always to turn in the direction indi- 
cated by the arrow. 




Escapements. 63 

A portion of this wheel is embraced by an anchor, A C B, 
centred at C, the extreme ends of which are formed into pallets, 
A m and B n : these pallets FIG. 63. 

may be flat or slightly convex, 
but they are subject to the 
condition that the perpendicu- 
lar to A m shall pass above C, 
and the perpendicular to B n 
shall pass between C and E. 
The point of a tooth is repre- 
sented as having escaped from 
the pallet B n after driving 
the anchor to the right hand; 
and the point g, by pressing 
against A ;//, is supposed to 
have already pushed the an- 
chor a little to the left hand, 
and thus the wheel can only proceed by causing a vibratory motion 
in the anchor, A C B. 

If the escape wheel engaged only a light metal anchor, such as 
that shown in the drawing, the rapidity of the vibration set up, as 
above described, would be very great ; but in a clock the object is 
to provide that the wheel shall advance by slow and regular steps, 
and the anchor is therefore controlled by the inertia of a com- 
paratively heavy swinging pendulum. 

There is one uniform method of connecting the anchor ancj 
the pendulum which can be seen in any clock. The pendulum, 
consisting often of a compound metal rod with a heavy bob, is 
swung by a piece of flat steel spring, and vibrates in a vertical 
plane a little behind that in which the anchor oscillates. To the 
centre of the anchor is attached a light vertical rod, having the end 
bent into a horizontal position, and terminating in a fork which 
embraces tl e pendulum rod. It follows that the anchor and the 
pendulum swing together as one piece, although each has a sepa- 
rate point of suspension. 

The same recoil is experienced upon each swing of the pen- 
dulum as that which we noticed in the last article, and the con- 
trivance is commonly known as the Recoil Escapement. 



64 Elements of Mechanism. 

ART. 54. The exact character of the action which takes place 
between the pendulum of a clock and the scape wheel has been 
the subject of a long and interesting mathematical investigation, 
and before mentioning the results which have been arrived at, we 
may state in general terms the nature of the problem. 

The going part of a clock consists of a train of wheels tending 
to move under the action of a weight or spring : if the last wheel 
of the train were left to itself, it would spin round with great ve- 
locity, and we should fail in obtaining any measure of time. 

The escapement is one part of a contrivance for regulating the 
velocity of the train of wheels, but the escapement alone is not 
sufficient : we require further a vibrating body possessing inertia, 
the motion of which cannot be suddenly stopped or reversed. 

Such a body is found in the pendulum, and a very intricate 
mutual action exists between the pendulum and the scape wheel. 
The function of the pendulum is to regulate and determine the 
periods and amount of onward motion in the scape wheel, whereas 
the office of the wheel is to impart such an impulse to the pendu- 
lum at each period of this onward movement as may serve to 
maintain its swing unimpaired, and may cause it to move with the 
same mathematical precision which would characterise the vibra- 
tions of a body swinging in vacuo, and uninfluenced by any dis- 
turbing causes. 

It is remarkable that in the case of the ideal pendulum, where 
there is no artificial resistance and no friction, the movement is in 
theory perpetual. In the case of the rigid pendulum the friction 
at the point of suspension and the resistance of the air would gra- 
dually diminish the arc of swing, and the movement would slowly 
subside and die away, although it might be many hours before it 
became quite imperceptible. The mechanism of a clock must 
therefore so act upon the pendulum as to maintain its swing un- 
impaired by these resistances, and it should be borne in mind that 
the swing of the pendulum is identical with that of the anchor in 
the last-mentioned escapement, whence it follows that any impulse 
or check given to the anchor is felt at once as an impulse or check 
upon the pendulum. 

Refer now to the recoil escapement described in Art. 53, and 
conceive that the escape wheel is urged onwards in the direction 



The Pendulum. 65 

of the arrow by the force of the clock train, so as to press its teeth 
slightly against the pallets of the anchor, the pendulum being hung 
from its point of suspension by a thin strip of steel, and vibrating 
with the anchor in the manner already stated. 

Let the arc A E C D B be taken to repre- 
sent the arc of swing of the centre of the bob 
of the pendulum. 

As the pendulum moves from B to E the 

point q of the escape wheel rests upon the oblique surface A m of 
the pallet, and presses the pendulum onward until the point of the 
tooth escapes at the end of the pallet. For an instant the escape 
wheel is free, and tries to fly round, but a tooth is caught at once 
upon the opposite side by the oblique edge B n, and the escape 
wheel then presses against the pendulum and tends to stop it, 
until finally the pendulum comes to rest at the point A, and com- 
mences the return swing. 

What now has been the action ? From B to E the force of 
the train as existing in the escape wheel has been acting with the 
pendulum and has performed its proper office in assisting to main- 
tain the swing ; whereas from E to A this force has acted against 
the pendulum. 

So also on the return swing, the escape wheel will ac f with the 
pendulum from A to D, and against it from D to B. 

The action then is alternately with and against the pendulum, 
and it might be supposed that the injurious effect of a pressure 
against the pendulum would be entirely corrected by the main- 
taining force in the other part of the swing ; but this is not the 
case. The pendulum no longer moves with what we may call its 
natural swing, as a free pendulum would oscillate, and any varia- 
tion in the maintaining force will disturb the rate of the clock. 

The matter has been carefully analysed by mathematicians, 
and they have shown that the principle of this escapement is radi- 
cally bad, because it is impossible to remedy entirely the harm which 
is done by continually interfering with the swing of the pendulum. 

There occurs also the useless expenditure of energy. It is 
almost superfluous to remark that no mechanical arrangement will 
ever bear a close scrutiny when it is so constructed as to throw 
away work. 

F 



66 



Elements of Mechanism. 



ART. 55. The dead-beat escapement W3& invented by Graham, 
and at once removes this primary objection. It is, however, most 
worthy of note that the change in construction which abolishes 
the defects due to the recoil, and gives the astronomer an almost 
perfect clock, separates the combination entirely from its original 
conception, viz., that of an apparatus for converting circular into 
reciprocating motion. No such conversion can be effected by 
Graham's escapement. 

The improvement is made clear by the sketch, and the student 
will observe that the pallet A has its lower edge in the form of a 

FIG. 65. 





circular arc, Ag, whose centre is C, and again that the upper por- 
tion of the pallet B is also a circular arc struck about the same 
centre. The oblique surfaces gm, np complete the pallets. Take 
the case shown in the diagram, which is enlarged so as to make 
the action more apparent. As long as the tooth is resting on the 
circular portion nr of the pallet, the pendulum is free to move, 
and the escape wheel is locked. Hence in the portion EA, and 
back again through AE, there is no action against the pendulum 
except the very minute friction which takes place between the 
tooth of the escape wheel and the surface of the pallet. Through 



Dead-beat Escapement. 67 

a space ECD the point of the escape wheel is pressing against the 
oblique edge np and is urging the pendulum forward. 

Then at D the tooth upon the opposite side falls upon A^, and 
the escape wheel is locked ; from D to B, and back again to D, 
there is the same friction which acted through EA or AE ; whereas 
from D to E the point of a tooth presses upon qm and urges the 
pendulum onward ; at E another tooth is locked upon the pallet 
B, and thus the action is reproduced in the order in which it has 
been described. 

It follows that any action against the pendulum is eliminated, 
or, more correctly, is rendered as nearly as possible harmless, and 
the difference between the ' recoil ' and the ' dead beat ' will be 
understood upon contrasting the three enlarged diagrams, which 
sufficiently explain themselves, the lower sketch referring to the 
recoil escapement. 

The term ' dead beat ' has been applied because the seconds' 
hand which is fitted to the escape wheel stops so completely when 
the tooth falls upon the circular portion nr. There is none of that 
recoil or subsequent trembling which occurs when a tooth falls 
upon B;z and is driven back. 

The actual construction of the dead-beat escapement having 
been explained, it only remains for us to state two of the principal 
conclusions which follow from a theoretical inquiry into the mo- 
tion of the pendulum. 

1. All action against the pendulum should be avoided, and if 
some such action be inevitable, it should at any rate be reduced 
to the smallest amount that is practicable. 

2. The maintaining force should act as directly as possible, 
and the impulse should be given through an arc which is bisected 
by the middle point of the swing. 

That is, the arc of impulse DCE should be bisected at the 
lowest point C. 

This latter condition cannot be exactly fulfilled, because the 
point of the escape wheel must fall a little beyond the inclined 
slope of the pallet in order that it may be locked with certainty. 

ART. 56. In a printing telegraph instrument the recoil escape- 
ment has been employed to control the rapidity of motion in a 
train ot wheels, and the number of vibrations of the anchor are 



68 



Elements of Mechanism. 



appreciated by listening to the musical note which it imparts to a 
vibrating spring. 

The anchor ACB (fig. 66) is centred at C, and vibrates rapidly 
as the scape wheel E revolves ; a strip of metal, F, carries on the 
oscillation to a steel spring which gives the note, and the velocity 
of the train can be regulated by an adjustable weight attached to 
the spring. 

Again, the same escapement forms part of the mechanism of 
an Alarum dock, where a hammer is attached by a bar to the an- 
chor, and blows are struck upon the bell of the clock in rapid suc- 
cession as the scape wheel runs round. 

FIG. 66 FIG. 67. 





ART. 57. The teeth of the wheel in an anchor escapement 
are sometimes replaced by pins, in which case the form of the 
anchor may be so altered that the action shall take place upon one 
side of the wheel, as shown in fig. 67. 

ART. 58. Circular may be converted into reciprocating mo- 
tion by the aid of cams. 

The term ' cam ' is applied to a curved plate or groove which 
communicates motion to another piece by the action of its curved 
edge. 

Such a plate is shown in fig. 68, and, as an illustration, we 
shall suppose that the portions al>, ca are any given curves, and 
that be is a portion of a circle described about the centre of 
motion. 

It is easy to understand that as the cam rotates in the direc- 



Cams. 



69 




tion of the arrow, the roller P at the end of the lever AP will be 

raised gradually by the curved 

portion ab, will be held at rest 

while be passes underneath it, 

and, finally, will be allowed to fall 

by the action of ca. 

In this way a cam may be 
made to impart any required mo- 
tion, and may reproduce in ma- 
chinery those delicate and rapid 
movements which would otherwise demand the highest effort of 
skill from a practised workman. 

ART. 59. The circular motion being uniform, the recipro- 
cating piece may also move uniformly, or its velocity may be varied 
at pleasure. 

i. Suppose that the reciprocating piece is a sliding bar, whose 
direction passes through the centre of motion of the cam-plate. 
Take C as this centre, let BP represent the 
sliding bar, and let A be the commencement 
of the curve of the cam -plate. 

The curve AP may be set out in the fol- 
lowing manner. 

With centre C and radius CA describe a 
circle, and let BP produced meet its circum- 
ference in the point R. 

Divide AR into a number of equal arcs 
Aa, ab, be, &c. 

Join Ca, C, O, &c., and produce them 
to/, q, r, &c., making ap, bq, cr, &c., respec- 
tively equal to the desired movements of BP pi 
in the corresponding positions of the cam- 
plate. The curve hpqr . . . P will represent 
the curve required. 

This curve will often present in practice 
a very irregular shape, but in the particular 
case where the motion of PB is required 
to be uniform, it assumes a regular and well-known form. 

Let CA = a, CP = r, PCA = 0, and let BP move in such a 




Elements of Mechanism. 



FIG. 70. 



manner that the linear velocity of P shall be constantly m times 
that of the point A, in other words, let RP = m . RA. 
Now RP = r a, and RA = ad. 

.'.r a = m . ad, 

which is the equation to the spiral of Archimedes. 
2. We will next examine the case where the centre of motion 
of the cam-plate lies upon one side of the direction of the sliding 
bar, and we shall find that the method of setting out the curve 
changes accordingly. 

Suppose that the direction of BP passes upon one side of the 
centre of motion C, draw CR perpendicular to BP produced, de- 
scribe a circle of radius CR, and conceive 
the motion to begin when A coincides with R. 
As a matter of theory such an extreme 
case is possible, and we will imagine it to 
exist in order to obtain the equation which 
represents the complete curve. Practically, 
the cam would be more effective in straining 
the bar than in moving it when the point P 
was near to the point R. 

Divide AR into the equal intervals A a, 
afr, be, &c., but now draw ap, bq, cr, &c., 
tangents to the circle, and equal in length 
respectively to the desired movements of BP 
during the corresponding periods of motion of the cam-plate. 

The curve Kpqr . . . P will be that required, and the analytical 
representation of it is the following : 

Let CP=r, CA=a, ACP=0, PCR=<p. 
The curve AP is now of such a character 
that the linear velocity of A shall be con- 
stantly m times that of P, or, in other words, 
RA=w . RP. 

But RA = a(6 + </>), RP = a tan 0, 
.*. a (6 + /)) = ma . tan 0. 




FIG. 7 




Now cos = -, and tan </> = 



tan 



V:;- 



COS 2 



Cams. 



whence + cos ~ l - 



Cor. Let m=i, or RA=RP, which would happen if AP were 
a stretched string unwound from the circle. The curve traced 
out by the end P of the string becomes in this case a well-known 
curve called the involute of the circle, and our equation takes the 
form 



which is the equation to the involute of a circle. 

ART. 60. The heart wheel has been much used in machinery, 
and is formed by the union of two similar and equal cams of the 
character discussed in the first part of Art. 59. 

A curved plate, C, shaped like a heart, actuates a roller, P, 
which is placed at the end of a sliding bar, or which may be at- 
tached to a lever PAB, 
centred at some point 
A, and connected by a 
rod BD to the recipro- 
cating piece. The pe- 
culiar form of the cam 
allows it to perform 
complete revolutions, 
and to cause an alter- 
nate ascent or descent 
of the roller P with a 
velocity which may be made quite uniform. 

Since a cam of this kind will only drive in one direction, the 
follower must be pressed against the curve by the reverse action of 
a weight or spring. 

ART. 6 1. In order to illustrate in a lecture the power of cams 
to produce any required movement, the late Professor Cowper ar- 
ranged a model which would write the letters R I, selected pro- 
bably as a compliment to the Royal Institution. 

The principle of this combination of cams will be readily under- 
stood if we remember that the successive movements of a point in 
directions parallel to two intersecting lines will suffice to enable 
the point to take up any position in the plane of the lines. 




.72 



Elements of Mechanism. 



The bars shown in the drawing have fixed centres at A and B, 
and it is apparent that if we were to remove the cam and fasten 
the joint R to the plane, we should be able to give P a vertical 
movement by the swing of the arms AE and RD. In the same 
way, if we fastened E to the plane and liberated R, the arm EP 
could swing about E, and P would then describe a small circular 
arc which would closely approach to a horizontal line. 

FIG. 73 





Connect now the bars with the cam as in the sketch, and press 
the pointers at Q and R against the curves of the respective cams. 
Let these cams revolve slowly about a centre, C (marked as a 
round spot in the smaller cam-plate), in the direction shown by 
the arrow, and the required letters will be traced out by the pencil 
at P. 

In the figure the letter R has just been completed, and the 
pencil is about to trace out the lower tail of the letter I. 

The two darkened lines in the cams are arcs of circles about 
the centre of motion where nothing is being done, the pencil re- 
maining at rest while the cam rotates through a small angle. 

This example shows us that a combination of two cam-plates 
actuating a simple framework of levers will give the command of 
any movement in a plane perpendicular to the axis of rotation. 
We shall presently see how to obtain a motion parallel to the same 
axis, and thus we can secure any required movement in space. 



Compound Cams. 



73 



ART. 62. In like manner two simultaneous rectilinear move- 
ments in lines at right angles to each other are obtained in a well- 
known form of sewing machine by the operation of grooved cams 
upon the face of a plate. 

The sketch is taken from a lecture diagram. 

1. The needle bar a, carrying the needle N, is constrained to 
move up and down in a vertical line between guides: it is driven 
by the lever BDP, which has a centre of motion at D, and is con- 
nected by a roller P with the groove marked nn in the cam-plate. 
The centre of motion of this plate is at C, and is marked by a 
strong dot. 

If the groove were formed in a circle round C the needle 
would remain at rest, and it receives its required motion under the 
constraint of the cam. The dotted lines show the extreme posi- 
tions of the lever during the motion. 

2. The shuttle S moves to and fro in the path or ' race ' be. 

FIG. 74. 




It is actuated by rod SV passing between guides df and ter- 
minating in a pin V which runs in the groove ;//. As the cam- 
plate rotates the pin V moves to and fro in a horizontal line, the 
extreme positions both of V and S being indicated by dotted lines 
in the diagram, and thus the needle and shuttle operate together 
in the manner required. The direction of rotation of the cam- 
plate is marked by an arrow. 

ART. 63. The movement of the needle in a sewing machine 



74 



Elements of Mechanism. 



is sometimes obtained from a peculiar form of cam, of the type of 
the heart wheel described in Art. 60. 

The peculiarity consists in driving the cam by a pin P, placed 
on the face of the circular plate. In the ordinary heart wheel the 
pin or roller, P, moves up and down while the heart-shaped piece 
rotates upon its centre. Here the heart-shaped piece moves up 
and down while the pin rotates. The result is that we find only 
FIG 75 * ne a P ex f the heart instead of the complete 

outline. The contrivance is well worth study- 
ing as an example of the combination of move- 
ments. 

The drawing shows the needle bar AB, 
carrying a needle at N, and guided by openings 
in the fixed frames aa and// 

The bar is attached to the darkly-tinted 
heart-shaped piece connected with it at the boss 
marked C, and the result is that the heart and 
the needle bar move as one thing. The pin P 
is attached to the circular plate, which rotates 
about its centre of figure, and it is clear that the 
needle bar is now nearly at rest in its lowest 
position, and will remain so until P gets round 
to the upper edge of the heart. The bar will 
then rise, and continue to rise while P is appar- 
ently descending the sloping side into the posi- 
tion shown by the dotted lines. The needle bar will then have 
reached its highest position, and will afterwards descend. 

ART. 64. In the striking part of a large clock the hammer 
may be raised by a cam, and may then be suffered to fall abruptly. 
The figure represents the cam devised 
for the Westminster clock ; the hammer 
rises and falls with the lever, AC, and 
the cam is so formed that its action com- 
mences at the extremity of the lever, and 
never departs sensibly from the same 
point ; the cam, al>, is a circle whose centre 
is at the point of intersection of the tangents 
to the rim of the wheel at a and c. 




FIG. 76. 




Punching Machine. 



75 



ART. 65. The lever punching machine is worked by a cam 
resembling that which we give as an example by Mr. Fletcher, of 
Manchester. The cam is here shown attached to the axis of the 
driving wheel, and the lever, which carries the punch in a slide 
connected with its shorter arm, is centred on the pin at E. 

FIG. 77. 




The curve of the cam is adapted to raising the longer arm of 
the lever bar in of a revolution of the driving shaft, it allows the 
lever to fall in the next 5 of a revolution, and finally leaves the 
punch raised, as shown in the sketch, during the remaining 2 of 
a revolution, thereby giving the workman an interval of time for 
adjusting the plate of iron before the next hole is punched. 

That this action occurs will be quite evident upon inspecting 
the form of the cam, and it will also be seen that the cam is pro- 
vided- with a circular roller B, which determines the form of the 
driving surface while the work is being done, and which is merely 
an arrangement for lessening the friction just at the time when the 
greatest pressure is being exerted. 

ART. 66. Cams are employed when it is required to effect a 
movement with extreme precision. Thus in a now obsolete ma- 
chine of Mr. Applegath for printing newspapers, the sheet of paper 



Elements of Mechanism. 



used to start upon its journey to meet the type at a particular in- 
stant of time ; an error of one-twelfth of a second would cause 
the impression to deviate half a foot from its correct position, 

and would throw two columns 
of letter-press off the sheet 
of paper. The accuracy with 
which the sheet was delivered 
was therefore very remarkable, 
and was insured by the assist- 
ance of the cam represented 
in the diagram (fig. 78) 

As C revolves, the roller at 
B drops into the hollow of the 
plate, thereby determining the fall of the lever AB, and by it the 
fall also of another roller which starts the paper upon its course 
to the printing cylinder. 

ART. 67. Hitherto we have considered the cam to be a plane 
curve or groove, but there is no such restriction as to its form in 
practice. Let us examine the following very simple case : 

FIG. 79. F 




A > 




CD is a rectangle with a slit RS cut through it obliquely: a pin 
P fixed to the sliding bar AB works in the slit. If the rectangle 
CD be moved in the direction RS, it will impart no motion to the 
bar AB ; but if it be moved in any other direction, the pin P will 
be pushed to the right or left, and a longitudinal movement will 
be communicated to the bar AB (fig. 79). 



Cams. 77 

The contrivance here sketched is of frequent use in some form 
or other, and we may point out its application in the rifling bars 
used at Woolwich in the manufacture of rifled guns. In work of 
this kind, where the greatest accuracy is demanded, the bore of 
the gun acts as a guide to the head of the rifling bar, and the 
cutter does its work while the bar is being twisted and pulled out 
of the gun. It is essential, therefore, to keep the cutter within 
the head while the bar is being inserted preparatory to the removal 
of a strip of the metal, and to bring it out again at the end of the 
stroke. 

In order to arrive at this result the bar is made hollow, and the 
tool-holder in the rifling head, shown in fig. 80, is made to move 
in and out laterally by means of a pin P working in an inclined 
slot, RS, in the internal feed rod. As the feed rod is pushed 
through a small definite space in either direction along the axis of 
the bar, the cutter will also move in or out in the direction of the 
dotted line AB. 

In discussing this motion there are two FIG. 81. 

cases to consider. R. 

1. Suppose that CD is moved at right 
angles to AB. 

Draw RN perpendicular to AB. A_ 

Th travel of CD = RN 
n travel of AB PN 

=tan RPN. 

2. Let CD move in a direction inclined at any given angle to 
the direction of the groove RS. 

Draw RN in this direction, and we 

, FIG. 82. 

have P 

travel of CD = RN 
travel of AB NP 

= sin RPN 

sin NRP' 

In other words, the velocity ratio of 
CD to AB is expressed by the fraction 
sin RPN 
sin NRP' a ta ^ es tne f rm tan RPN when the angles at R and 

P make up a right angle. 



Elements of Mechanism. 



ART. 68. Next let CD be wrapped round a cylinder ; it will 
form a screw-thread, and the revolution of the cylinder upon its 
FIG g axis will be equiva- 

A T , lent to a motion of 

the rectangle at right 
angles to the bar, in 
the manner shown in 
the preceding article. 
We shall have, therefore, by the arrangement in the figure, a 
continuous uniform rectilinear motion of the bar AB during the 
revolution of the cylinder upon which the screw thread is traced. 

If the pitch of the screw be constant, the motion of PB will 
be uniform, and any change of velocity may be introduced by a 
proper variation in the direction of the screw-thread. 

If the screw be changed into a circular ring, AB will not move 
at all. 

It is, then, a matter of indifference whether the cam be a 
groove traced upon a flat plate or a spiral helix running round a 
cylinder. In the first case motion ensues when the groove departs 
from the circular form, and the distance from the centre varies ; in 
the second case motion ensues the moment the groove deviates 
from the form of a ring, whose plane is perpendkular to the axis. 
As an illustration of a cam of the latter character, we may 
refer to the diagram, which shows a form very much used where 
a small motion of a lever is re- 
quired; the lever ACB is centred 
upon the point C, and will com- 
mence to move as soon as the 
pin at the end A reaches that 
portion of the ring which departs 
from the circular form. 
7V0&. This kind of cam has the property of giving a motion 
parallel to the axis upon which it is shaped. 

ART. 69. As, an example of two cam grooves in juxtaposition, 
one formed upon a flat plate and the other upon a cylinder, we 
may refer to a machine which was formerly used for shaping coni- 
cal box-wood plugs of the kind required for expanding an elon- 
gated bullet into the grooves of a rifle. 



FIG. 84. 




Cams. 



79 



Bullets of this class have a hollow recess at the base, into which 
a plug is fitted, and when the powder is fired the thin sides of the 
recess are forced into the grooves of the rifle by the action of the 
plug. At the present time these plugs are made of compressed 
clay, but formerly they were cut from a rod of box-wood by ma- 
chinery. 

The wood is first cut up into small square rods by means of 
circular saws, and one of the wooden rods is fixed in a saddle, 
as shown at H in fig. 85. Our drawing is taken from a lecture 
diagram by Sir J. Anderson. 

FIG. 85. 




The next operation is to bring a revolving cutter, which 
carves out the shape of the plug, upon the end of the rod H. 
This is effected by the lever PCE, centred at C, and having the 
end E jointed to a spindle which carries the cutter at its opposite 
end. A driving pulley M is rotated at a high velocity by means 
of a strap and pulley not shown in the drawing, and the longitu- 
dinal movement to and fro of the spindle, with its cutter and 
driving pulley, is effected by a grooved cam on the cylinder A. 
It will be apparent that this cam ought to be, and, in fact is, shaped 
precisely after the fashion of the cam in fig. 84. 



8o Elements of Mechanism. 

As soon as the cutter had been forced down so as to form the 
end of the plug, it was withdrawn, and the final operation was to 
cut off the shaped piece. In the earlier machines a circular saw 
made a transverse movement, to cut off the plug, whereby the 
plugs and the chips were mixed together in the same heap, and 
Sir J. Anderson has stated that the labour for their separation cost 
more than for their manufacture. But soon it was arranged that the 
whole saddle HR carrying the plug rod should be shifted bodily 
back through a small space against a circular saw S, whereby the 
plugs might be dropped into a separate box. 

The circular saw S is driven by a pulley N, and the trans- 
verse motion of HR is effected by the cam-plate B, which acts 
upon a roller at Q and connecting rod QT. The saddle HR is 
placed upon the top of a weighted rocking frame, and it is apparent 
that the cam B would give the transverse movement required. 

D is a driving spur wheel which gives motion to the cam shaft. 
There is also a contrivance which we do not explain for pushing 
the rod H forward by the length of a plug at the end of each 
stroke. 

ART. 70. We subjoin a further example, devised many years 
ago, in which a reciprocating movement is imparted to a frisket 
frame in printing machinery, and it will be presently seen that 
the required result can be obtained in a much more simple 
manner. 

The use of a cam-plate allows of an interval of rest at each end 
of the motion, and enables the printer to obtain an impression, 
and to place a fresh sheet of paper upon the form. 

Here AH is the reciprocating frame attached to the combina- 
tion of levers GFEDCB by the link AB (fig. 86). 

At the end of the lever, FG, is a sliding pin which travels 
along the grooves in the flat plate centred at O, and determines, by 
its position, the angular motion of the levers about the fixed centres 
at F and C. 

Where the groove is circular, which occurs in those portions 
which are to the left hand of the vertical dotted line, the levers 
remain at rest, and they change into the position shown by the 
dotted lines when the sliding pin passes from the outer to the 
inner channel. The pin is elongated in form, as shown at G', 



Cams. 



8 1 



and is thus capable of passing across the intersections of the 
groove. 





Precisely the same character of movement may be obtained 
by the aid of a helical groove traced upon a revolving drum. 
The intervals of rest occur when the groove assumes the form 
of a flat ring, whose plane is perpendicular to the axis of the 
drum. 

A right and left-handed screw-thread is traced upon the worm 
barrel, AB. which revolves 

Pm RT 

in one uniform direction 
a pin attached to the table 
of a printing machine fol- 
lows the path of the groove 
upon the barrel, and its 
form is elongated so as to 
enable it to pass in the right direction at the points where the 
grooves intersect. 

The interval of rest commences with the entry of the pin into 
the flat ring at either end of the barrel, and may be made to 
occupy the whole or any part of a revolution of AB, according as 
the grooves enter and leave the ring at the same or different 
points. 

This construction dispenses with the complicated system of 
levers, which constitutes such a serious defect in the other ar- 
rangement. 

Mr Napier has patented an invention which causes the in- 

G 



32 



Elements of Mechanism. 






Fig. i. 



terval of ' rest ' to endure beyond the period of one revolution of 
the barrel. 

At the entrance to the circular portion of the groove a mov- 
able switch is placed, and it is provided that the switch shall be 
capable of twisting a little in either direction upon its point of 
support, and also that the pin upon which the switch rests shall 

admit of a small longi- 
tudinal movement pa- 
rallel to the axis of the 
barrel, the pin itself be- 
ing urged constantly to 
the right hand by the 
action of a spring. 

In fig. i the shuttle 
is seen entering the cir- 
cular portion of the 
groove, and twisting the 
Fig- 3- switch into a position 
which will allow the shuttle to meet it again, as in fig. 2, and to 
make a second journey round the circular ring. 

The spring which presses the point of support of the switch to 
the right hand will now cause it to twist by means of the reaction 
which the passing pin affords, and the consequence will be, that 
the switch will be left in the position shown in fig. 3, and will 
guide the shuttle into the helical portion of the groove. Thus the 
p'eriod of rest will be that due to about one and two-thirds of a re- 
volution of the barrel. 

ART. 71. We remark, in conclusion, that when the mechanic 
causes the moving body to be influenced by a pin which exactly 
fits the groove along which it travels, it is obvious that the moving 
body will take the exact position determined by the pin ; on the 
other hand, where the cam is merely a curved plate pushing a 
body before it, there is no certainty that this body will return 
unless it be brought back by a weight or spring. Hence it arises 
that double cams have sometimes been employed in machinery, 
and we take the next example from an early form of power- loom. 
AB is the treadle, E and F are the cam-wheels or tappets, 
which revolve in the directions shown by the arrows, and in such 



Cams. 



relative positions that the projections and hollows are always 
exactly opposite to each other. As the cams rotate, the treadle, 
AB, is alternately elevated and depressed, and the threads of the 



FIG. 89. 




FIG. 90. 



warp are opened so as to permit the throw of the shuttle during 
the operation of weaving the fabric. 

ART. 72. Cams are frequently employed for the purpose of 
opening or closing with rapidity the valves of a steam cylinder, or 
other valves concerned in what is termed the expansive working of 
steam, that is, the cutting off the supply of steam before the end 
of the stroke of the piston. 

In a movement of this kind the cam is re- 
quired to lift the valve rapidly from its position 
of rest, then to hold it up for a time while the 
steam is passing through, and next to allow it 
to drop into its seat and remain at rest. 

The cam usually operates upon one end of 
a lever, the other end of which is connected 
with the valve, and it is apparent that it will 
suffice to surround the crank shaft of the engine by a plate or 

QZ 




Elements of Mechanism. 



cylinder having a circular portion ef, on which the end of the 
valve lever rests when the valve is closed, and a raised portion, 

AB, also circular, upon which the end of the valve lever runs when 
the valve is to be opened. Thus there are two circular portions 
which determine the opening and closing of the valve, and an 
arbitrary sloping portion connects AB with ef, and determines the 
rapidity with which the changes take place. 

For some purposes, as where steam is to be expanded in vary- 
ing degrees, the raised portions are of different lengths, as AB, 

AC, AD, arranged in successive steps, one behind the other, 
whereby the valve may be held open for different periods. 

Also, it is manifest that the cam may lie on the face of the plate 
instead of being part of its edge, and that in effect two portions of 
flat plates rotating about a common axis perpendicular to each, 
and raised one above the other, with a sloping surface connecting 
them, would be a mechanical equivalent for the cam described. 
Such a cam-plate was used by Sir W. Fairbairn. 

ART. 73. Where the cam- plate is required to effect more 
than one double oscillation of the 
sliding bar during each revolution, 
its edge must be formed into a cor- 
responding number of waves. 

There is an example in telegraph 
commutators, the interruptions of 
the current being caused by the 
vibrations of a lever, PCQ, centred 
at C, and whose angular position is 
determined by a pin travelling in the 
groove. 

As the wheel revolves, it can im- 
press any given number of double 
oscillations upon the lever. 
ART. 74. We have hitherto confined our attention to simple 
examples in the geometry of motion : we shall now extend our 
view of the subject, and shall consider the communication of 
motion when the driver is a toothed wheel or pulley rotating con- 
tinuously upon a fixed centre or axis ; and in order to generalise 
still further, we shall suppose the reciprocating motion to be either 



FIG. 91. 




Reversing Motion. 8 5 

rectilinear or circular. In this manner we shall be enabled to 
bring under one point of view a great variety of useful mechanical 
contrivances. 

The student will be aware that in the transfer of force by 
machinery, the moving power is carried from one piece of 
shafting to another, throughout the whole length and breadth of 
the factory ; it passes from point to point, enters each separate 
machine, and gives movement to all the several parts which may 
be prepared for its reception. 

Now it must be remembered that the engine is never reversed, 
and that the power continues to flow onward in one uniform 
direction. 

Take the case of a machine for planing iron : here the princi- 
pal movement is that of a heavy table sliding forwards and back- 
wards, and carrying the piece of metal which is the subject of the 
operation. 

There are two methods of obtaining the desired result : the 
power may be poured, as it were, into the machine by a stream 
running always in one direction, and the reciprocation may be 
provided for by the construction of the internal parts, or the flow 
of the stream may be reversed by some intermediate arrangement 
external to the machine itself. 

ART. 75. The former method is that usually adopted, and 
we shall now examine those machines where the reciprocation de- 
pends upon the internal construction of the moving parts. 

And, first, we shall discuss a very simple and useful reversing 
motion which is obtained by a combination of two or three spur 
wheels, and which depends upon an obvious fact. 

FIG. 92. 




Let A,B,C represent three spur wheels in gear ; it will be seen 
that A and B turn in opposite directions, while A and C turn in 
the same direction. If then we connect two parallel axes by a 



86 



Elements of Mechanism. 



combination of two and three spur wheels alternately, and pro- 
perly arrange our driving pulleys, so that the power shall travel 
first through one combination and then through the other, we 
shall have a movement which has been adopted by Collier in his 
planing machines, and which has been subsequently much used by 
other makers. 

The power is now derived from the shafting by means of a 
band passing over a drum on the main shaft and over one of the 
three pulleys, E, I, F, at the entrance into the machine. 

Of these pulleys E is keyed to the shaft, I rides loose upon it, 
while F is attached to a pipe or hollow shaft, through which the 
shaft connecting E with A' passes, and which terminates in the 
driving wheel A. 

FIG. 93. 




There is also a second shaft B'C, which carries the toothed 
wheels B' and C. 

B is an intermediate wheel riding upon a separate stud. 

When the band drives the pulley E, it is clear that A' and B' 
turn in opposite directions ; whereas the motion is reversed when 
the band is shifted to F, for in that case A and C turn in the same 
direction. When the driving band is placed upon I, the machine 
remains at rest. 



Reversing Motion. 87 

The rotation of B'C may be made much more rapid in one 
direction than in the other, and the construction is therefore par- 
ticularly useful in machinery for cutting metals. 

The slow movement occurs while the cutting tool is removing 
a slip of metal, and the return brings the table rapidly back into 
the position suitable for a new cut. 

ART. 76. This contrivance is in common use, and the draw- 
ing is from a machine arranged for cutting a screw-thread in the 
interior of the breech of an Armstrong gun. 

In this case the driving pulleys are placed between the wheels 
A and A', and are formed in such a manner that the pulley F and 
the wheel A make one piece, and ride loose upon the shaft HK, 
as do also, in their turn, the pulley E and the wheel A' : the 
wheel M is keyed to HK, so as to rotate with it, and is further 
attached by a coupling to the muzzle of the gun which is to be 
operated upon. 

FIG 94 




When the strap is upon E, the motion travels from A' to B', 
and so on to L and M, causing the gun and the shaft HK to 
rotate together slowly in one direction ; whereas, upon shifting the 
strap to F, the motion passes from A to C through a small inter- 
mediate wheel, and thence to L and M, whereby the rotation of 
the gun is reversed, and a higher speed is introduced. 



88 Elements of Mechanism. 

The object of the machine is to copy upon the interior of 
the breech of the gun a screw-thread which is formed upon the 
end R of the shaft HK. 

For this purpose the shaft HK is screwed, as shown, and 
a slide-rest carrying a cutter advances longitudinally along the 
gun, with a motion derived directly from a nut which travels along 
the screw-thread formed upon R. Since the cutter can only re- 
move the metal while passing in one direction, there is a loss of 
time during the return motion, which it is the object of this com- 
bination to reduce as much as possible. 

ART. 77. The same combination, slightly modified, is adopted 
generally in planing machines, and is valuable by reason of the 
uniformity of the movement, the rate of advance of the table being 
perfectly constant. 

It also possesses the important advantage of causing the table 
to traverse with a quick return movement when the cutter is not 
in action. 

We give so much of the machine as will explain the method of 
reversing the motion of the table. When the strap is upon the 
pulley F, the wheel A turns in one direction. When the strap is 

FIG. 93. 




upon the pulley E, the motion passes to B, which turns with E, 
and thus the axis, CD, is made to revolve in the opposite direction 
with a reduced velocity. 



Quick Return Movement. 



89 



The wheels A and D both engage with another wheel which 
actuates the table, and the reversal takes place when the moving 
power is transferred from the wheel A to D; but inasmuch as it 
would be difficult to give a clear representation of the movement 
without a sketch in perspective, an additional lecture diagram has 
been prepared wherein the toothed wheels are represented by 
circular discs with smooth edges. 

FIG. 96. 




Taking the two diagrams together, it will be apparent that the 
pulleys, E, I, and F, are arranged by the side of the first toothed 
wheel, A, whereby either F drives A, or E drives B, C, and D. 

It has been stated that the wheels A and D both engage with 
another wheel which actuates the table, and the second drawing 
shows this additional wheel, H, engaging both with A and D on 
one side, and connected directly with a compound or stepped 
wheel, K, which is placed under the rack R. 

The object being merely to indicate the position and arrange- 
ment of the working parts, the long rack which lies underneath the 
whole bed of the table is here marked as a short piece R rolling 
upon the disc K. 

ART. 78. We shall now examine another class of reversing 
motion's, and shall commence in the most elementary manner. 

Conceive a disc E, having a flat edge, to run between two 
parallel bars, AB and CD, arranged in a rectangular frame ; and 



9 o 



Elements of MecJiamsm. 



conceive, further, that the frame can be raised or depressed so as 
to bring AB and CD alternately into contact with E (fig. 97). 

If the disc rotates always in the direction of the arrows, it will 
move the frame to the left when brought into contact with CD, 
and to the right when brought into contact with AB. 

We have therefore a reversing motion within the limits of the 
frame. 

FIG. 97 FIG. 98. 





In order to make the motion continuous, it will only be ne- 
cessary to alter the bars into circular strips or discs, as shown in 
fig. 98, and we shall reverse the motion of the vertical axis by 
bringing the upper or lower discs AB and CD alternately into 
close contact with the driver E. 

In this way we obtain the first idea of a reversing motion, and 
it only remains for us to improve the general construction and 
arrangement of the working parts so as to make it practically 
useful. And we should observe that inasmuch as the rolling 
action of cones is more perfect than that of circular discs, for the 
reasons already explained^ in the introductory chapter, it will be 
better to substitute cones for the discs, in the manner shown in 
fig. 99, and the reversal will occur, just as before, when AB and 
CD are alternately brought into frictional contact with the driving 
cone E. 

The geometrical condition of rolling will demand that the 
vertex of the driving cone E shall coincide with that of AB in one 
position of contact, and with that of CD in the other ; hence the 
vertices of the two cones AB and CD must be separated through a 
small space equal to that through which the common axis is shifted. 

If we desire to transmit force beyond the limit at which the 
cones would begin to slip upon each other, we must put teeth 



Reversing Motion. 



upon the rolling surfaces, as in fig. 100, and we thus obtain a re- 
versing motion which has been used in spinning machinery. 



FIG. 99. 



FIG. ioo. 




Here a bevel wheel, E, is placed between two wheels, A and C, 
which are keyed to the shaft whose motion is to be reversed, the 
interval between A and C being enlarged so that E can only be in 
gear with one of these wheels at the same time ; the reversal is 
then effected by shifting the piece AC longitudinally, so as to 
allow E to engage with A and C alternately. 

ART. 79. A reversing motion which depends upon the shift- 
ing of wheels in and out of gear is not perfect as a piece of 
mechanism ; we must try, therefore, to convert it into another, so 
arranged as to give the reversal by passing a driving clutch from 
one wheel to the other, the wheels concerned in the movement 
remaining continually in gear and fixed in position. 

For this purpose we employ one working pulley F, keyed upon 
the shaft ED ; by its side we place a second pulley I, which rides 
loose upon the shaft, and which carries the driving band when no 
work is being done. 

The wheels A and C ride 
loose upon the shaft ED, and 
the intention is to impart the 
motion of the shaft ED, which f 
is driven by steam power, 
to the wheels A and C alter- 
nately. 

We now fit upon the shaft 
E a sliding clutch N, having projections which serve to lock it to 




J" 1 



Q2 Elements of Mechanism, 

A or C as required, and we place also a projection, or feather, 
upon the inner part of the clutch which slides in a corresponding 
groove formed in the shaft, so that N must always turn with ED. 
It is clear, therefore, that if we allow the clutch N to engage with 
A we shall communicate to B a rotation in one direction, and 
that, further, we shall reverse the rotation of B if we connect C 
with N, for the student will see that in this combination A and C 
must always rotate in opposite directions, and that the rotation of 
B as derived from A must be different from that which B would 
derive from C. 

This reversing motion may be commonly seen in steam cranes. 
The shaft ED is then driven directly by a steam-engine attached 
to the crane, and the sliding clutch may be locked to either bevel 
wheel by a friction cone, and is pushed to the right or left by 
means of a lever which grasps it without preventing its rotation. 

There is another application in screwing machines where a 
rapid reversal is required. In this case the ,;haft ED is reversed 
by the action of the bevel wheels, instead of imparting its rotation 
to each of them in turn. The driving pulley F being attached by 
a pipe to the wheel A, the reversal is effected by shifting the 
clutch, and thereby locking the shaft ED to the wheels A and C 
alternately. 

ART. 80. We have next to examine the application of this 
reversing motion in planing machines, and shall describe the com- 
bination of three pulleys with three bevel wheels which has been 
adopted by Sir J. Whitworth. 

In pursuing an inquiry into machinery of this character we 
may remark that the principle of machine copying, whereby a 
form contained in the apparatus itself is directly transferred to the 
material to be operated upon, is the distinguishing feature of all 
planing machines. The application of this principle is perfectly 
general, and, as a rule, wherever a process of shaping or moulding 
is well and cheaply performed by the aid of machinery, we find 
that some skilful and carefully arranged contrivance for transfer- 
ring a definite form is contained within the machine. 

In the earliest form of planing machine, a method of carrying 
the cutter along parallel bars was adopted, and the present practice 
is to employ perfectly level and plane surfaces called Vs, which are 



Reversing Motion. 



93 



placed on either side of the machines, and are shaped exactly as 
their name indicates ; their form gives a support to the table, pre- 
vents any lateral motion, and allows the oil required for lubrica- 
tion to remain in a groove at the bottom, from whence it may be 
worked up by the action of the machine. The table has projecting 
and similar Vs which rest upon the former, and the object of the 
mechanism is primarily to cause the table to move in either direc- 
tion along the grooves, and thus to copy upon a piece of iron 
supported thereon, and carefully bolted down, an exact plane sur- 
face which possesses the truth of the guiding planes. 

Whether it may be better to move the table by a rack and 
pinion or by a screw is a subject upon which different opinions 
are held, and at all events the quick return movement which is 
given by a combination of spur wheels, as already described in 
Art 77, is extremely valuable. 

To recur to Sir J. Whitworth's arrangement, we find that he 
effects the required movement by rotating a screw which runs 
along the central line of the bed, and which imparts to the table a 
perfectly smooth traversing motion, equal of course in exactness, 
if not superior, to that which could be obtained by the best-con- 
structed wheelwork. 




There are now three pulleys, E, I, and F, whereof I is an idle 
pulley, and rides loose upon the shaft ; E is keyed to a shaft ter- 
minating in the bevel wheel C, and F fits upon a pipe through 
which the shaft connecting E and C passes, and which terminates 
in the bevel wheel A. 

B is a bevel wheel at the end of the shaft whose direction of 
rotation is to be reversed. 



94 



Elements of Mechanism. 



It is clear that the motion of the wheel B is reversed when the 
driving strap is shifted from E to F. 

One objection to this movement consists in the fact that it 
does not permit the motion of B to be more rapid in one direction 
than in the other, and in order to economise the steam-power to 
the fullest extent, a method of rotating the tool-box was adopted 
by which means the cut was made while the table traversed in 
either direction. This reversal answers very well in planing ordi- 
nary flat surfaces. 

It may, however, be so arranged as to obtain a quick return by 
making A and C of equal size, and by causing them to gear re- 
spectively with two unequal wheels upon the axis of B, or a com- 
bination of spur and bevel wheels may be employed. 

ART. 8 1. The contrivance just described is shown in fig. 103, 




as applied in a machine for rifling guns, and the method adopted 
is precisely that so generally employed in planing machines. 

The three pulleys and the three bevel wheels are connected to- 



Reversing Motion. 95 

Aether in the manner already indicated, and the bevel wheel B, by 
its rotation, causes a saddle S carrying the rifling bar to move 
along the screw in the direction of its length. A bell crank lever, 
MLN, controls the bar PQ, which carries a fork used to shift the 
strap, the arms of the lever lying in different horizontal planes, 
while a movable piece, R, fixed at any required point of the bar, 
NR, is caught by a projection on the saddle as it passes to the 
right hand, and thus the bell crank lever is actuated, and the strap 
is carried along from E to F. 

A weight falls over when this is taking place, and gives the 
motion with sharpness and decision, so as to prevent the strap 
from resting upon I during its passage. On the return of the 
saddle to the other end of its path, a similar projection again 
catches a second piece upon the sliding bar NR, and the strap is 
thrown back from F to E. 

This bell crank lever, as employed for shifting the strap, is 
worthy of notice ; it consists of two arms, LN and LM, lying in 
different planes and standing out perpendicularly to an axis. It is 
a contrivance which affords a ready means of transferring a motion 
from one line, RN, to another, PMQ, which lies in a perpendicular 
direction at some little distance above it (Art. 36). 

ART, 82. Where the reciprocation is effected by a contrivance 
external to the machine, two driving bands may be employed : of 
these one is crossed, and the other is open, and it has been 
already pointed out that the followers will turn in opposite direc- 
tions, although they derive their motion from a single drum, 
which, being driven directly by the engine, must rotate always in 
one direction. 

The form which the arrangement assumes in practice is shown 
in the sketch, fig. 104, where PQ is the driving shaft carrying the 
drums R and S. 

Confining our attention at first to the left-hand diagram, we 
observe that one of the driving bands is represented as crossed, 
and that the rotation of the lower shaft HL is to be derived from 
each band alternately. 

There are three pulleys, whereof A and B are each loose upon 
the shaft, and are about twice as broad as C, which is a working 
pulley. 



96. 



Elements of Mechanism. 



The bands are shifted by two forks, and remain always at the 
same distance from each other. In the diagram the crossed strap 




is upon the idle pulley, and the open strap is on the working 
pulley, the result being that the shafts PQ and HL rotate in the 
same direction. When the bands are shifted a little to the left 
both straps will lie on the respective idle pulleys, A and B, and 
the shaft HL will cease to rotate. 

Whereas, upon shifting the straps still more to the left the 
crossed strap comes upon C, and the shaft HL begins again to 
rotate, but in the opposite direction to that in which it moved 
previously. 

Here the rate of rotation is the same in either direction, but it 
may be varied by connecting the drum S with two pairs of pulleys, 
viz., D, E and F, K of unequal size, upon the lower shaft NM. 
The extreme pulleys D and K are the working pulleys, and the 
reversal is effected just as before, but NM rotates more rapidly in 
one direction than in the other. 

ART. 83. There is yet another most ancient contrivance for 
changing circular into reciprocating motion, which will repay the 
trouble of analysing it. It is deduced from the same triangle as 
that concerned in the motion of the crank and connecting rod, 
but the varying dimensions of the sides are arrived at in a dif- 
ferent manner. One simple form is obtained when the points C 
and Q in the triangle CPQ become fixed centres of motion, the 



Quick Rettirn Motion. 



97 



FIG. 105. 



crank CP being less than CQ, and the extremity P of the crank 
CP moving in a slot or groove running along the line QR. 

The drawing shows an arm CP centred at C, and conveying 
motion to the grooved arm QR by means of a pin, P, which fits 
into the groove. As CP revolves with a uniform velocity, and in 
a direction opposite to the hands of a watch, it will cause QR to 
swing up and down to equal distances upon either side of the line 
QC, but with this peculiarity, that the upward swing will occupy 
less time than the down- 
ward swing. 

The motion of QR 
will be variable, its ve- 
locity changing at every 
instant, and we must 
endeavour, in the first instance, to discover an expression for its 
rate of motion as compared with that of CP. According to well- 
established rules, we estimate the relative rates of motion of two 
revolving pieces by comparing the sizes of the small angles de- 
scribed by either piece in a very minute interval of time reckoned 
from any given instant. 

ART. 84. Let now P/ represent the small arc described by P 
in a very minute interval of time, such as the -nnn^h P art f a 
second. Join Q/>, C/, and draw Pn perpendicular to Q/>. 

FIG. 106. 





Then angular ve! of QR . *& in the , imi 
angular vel. of CP L PC/ 



= limit of 



98 Elements of Mechanism. 

But ~Pn = P/ cos/P = P/ cos RPC = P/ cos (C + Q). 

angular vel. of QR_CP cos (C + Q) 
' ' angular vel. of CP ~~ PQ 

We may test this formula in the usual way ; for instance, let 
C + Q = 90, in which case QR touches the circle, then 

cos (C + Q) = cos 90 = o ; 
.*. angular vel. of QR=o, 

or QR stops, as we know it must do. 

Next, let C = o, Q = o, or let P be crossing the line CQ, then 
cos (C + Q)= cos. 0=1. 

. angular vel. of QR = CP 
' 'angular vel. of CP QP' 

or the vel. of QR is as much less than that of CP as QP is greater 
than CP, a result which is evidently true. 

If it be required to find the position of QP when P is at any 
given point of its path, we have the equation 
CP sin C 



whence the angle PQC is known in terms of C. 

If we draw QD, QH, tangents to the circle described by P, 
it will be evident that the times of oscillation of the arm will be 

FIG. 107. 




unequal, and will be in the same proportion as the lengths of the 
arcs DLH, DKH. 

The ratio of the angular velocities of QR and CP may also be 
obtained by analysis. 

Let CQP = fc PCQ = 0, CP = a, CQ = c, 



Quick Return Motion. 99 

.-.o=sm (0+0) cos ^~sin cos (9 + 0) j i +^J J 
/.sin cos (0 + 0) = {sin (0+0) cos 0-cos (0 + 0) sin 0}- 



ART. 85. Another way of looking at the motion, which is 
technically known as a slit-bar motion, is to consider that PQ is 
a line of indefinite length jointed at P to the crank CP, and con- 
strained to pass through an opening at the fixed point Q. As 
before, the condition that CP is less than CQ must maintain, or 
the line PQ will no longer oscillate. 

This again is the movement in an oscillating engine, where the 
steam cylinder is swung upon trunnions, and the crank CP is con- 
nected by a piston rod to a piston moving up and down in the 
cylinder. 

Whichever of the above forms may be selected as an illustration, 
the movement is precisely the same. 

FIG. 108. 




Taking CPQ as the triangle, draw CR perpendicular to QP 
produced, and draw PH parallel to CR. 

CP cos (0 + 0)_CP RP_RP_CH 
PQ PQ CP PQ HQ 

angular vel. of QR _ CH 
r ' angular vel. of CP HQ' 

Referring to the oscillating cylinder, let V be the linear velocity 
of the crank pin P, and v the velocity of the piston, which moves 
in a cylinder swinging on trunnions at Q, and which, for sim- 
plicity, we will indicate by the point S in the drawing. 



ioo Elements of Mechanism. 

Then the velocity of P resolved along PQ is equal to the 
velocity of S. 

..VsinRPC = z>, 

v CR 
r ' V = CP' 

which gives the velocity ratio of the piston to the crank pin in an 
oscillating engine. 

ART. 86. Hitherto we have supposed CP to be less than CQ, 
and the result has been that QR swings about the point Q in un- 
equal times ; but we will now arrange that CP shall be greater 
than CQ, in which case QR will sweep completely round with a 
circular but variable motion. We shall, in fact, have solved the 
problem of making a crank revolve in such a manner that one 
half of its revolution shall occupy less time than the other half. 

Now this is a very important result, and is of great value in 
machinery, because if the crank be made to perform its two half 
revolutions in unequal times, it follows 
that any piece connected with it by a link 
may be caused to advance slowly and re- 
turn more rapidly ; a movement which, as 
we have already pointed out, is peculiarly 
useful in machines for cutting metals. 

Constructing as before, let C be the 
centre of the circle described by P. Then 
the equation tan 




gives the position of QP when that of the crank is assigned. 

A] angular vel. of QR _ P_^ _._ P/ P/ cos CPQ CP 
S angular veL of CP ~~ PQ ' CP ~ .~PQ~~ P/ 



Cor. i. If CQ be small, the angle CPQ will be small also, and 
we shall have cos CPQ = i nearly ; in which case the angular vel. 

of QR varies as ~, while that of CP remains constant. 
Cor. 2. When CQP is a right angle, we have 



Quick Return Motion. 



101 



FIG. no. 



cosCPQ=, 

that is, the angular vel. of QR = the angular vel. of CP. 

This happens twice during a revo- 
lution, and gives the line of division of 
the inequalities of the motion of QR. 
Hence, if we draw DQH perpendicular 
to CQ, and cutting the circle described 
by P in the points D and H, the times 
of each half-revolution of QR will be 
in the proportion of the arcs DKH and 
DLH. 

Cor. 3. The angular ve 
locity ratio between QP and 
CP may be set out in a sim- 
ple geometrical form just as 
in the previous case. 

Draw PH perpendicular 
to QP, and QS parallel to 
PH. 




FIG. in. 




Q 



Then angular vel. of QP _ CP 
angular vel. of CP ~ PQ 

-CP x PQ 

PQ PS 

= CP 

PS 



QH 

ART. 87. If BR be made to carry a link, RQ, as in the case 
of the crank and connecting rod, the linear motion of Q will be 
the same in amount as if BR revolved uniformly, but the periods 
of each reciprocation will in general be different. (Fig. 112.) 

The difference in the times of oscillation will depend upon the 
direction of the line in which Q moves. 

The best position for that line is in a direction perpendicular 
to CB. We have shown that the times of oscillation are always as 
the arcs DKH and DLH, and it is also evident that the inequality 



102 



Elements of Mechanism. 



between these arcs is greatest when DH is perpendicular to CB, 
and diminishes to zero when DH passes through CB. 

We have now an arrangement very suitable for effecting a quick 
return of the cutter in a shaping machine. 

Let one end of a connecting rod be made to oscillate in a line 
perpendicular to CB, or nearly so, and let the crank BR be driven 
by an arm, CP, which revolves uniformly in the direction of the 
arrow, we at once perceive that Q will advance slowly and return 
quickly, the periods of advance and return being as the arcs DLH 
and DKH. 

FIG. 112. 




FIG. 113. 



ART. 88. Such a direct construction is not very convenient 
for the transmission of force, and it has been so modified by Sir 
J. Whitworth in his Shaping Machine, that the principle remains 
unchanged, while the details of the moving parts have undergone 
some transformation. 

This machine is analogous tp a planing machine, but there 
is no movable table ; the piece of metal to be shaped is fixed, 

and the cutter travels over it. 
The object is to economise 
time, and to bring the cutter 
rapidly back again after it has 
done its work. 

The arm CP is here ob- 
tained indirectly^ by fixing a 
pin, P, upon the face of a 
plate, F, which rides loose 
upon a shaft, C, and is driven 
by a pinion, E. 




Shaping Machine. 103 

As the wheel F revolves upon the shaft represented by the 
shaded circle, the pin moves round with it, and remains at a con- 
stant distance from its centre. 

A hole, B, is bored in the shaft, C, and serves as a centre of 
motion for a crank piece, DR, shown in fig. 114. The connecting 
rod, RQ, is attached to one side of this crank piece, and the pin, 
P, works in a groove upon the other side. Thus the rotation of 
the crank causes the end Q to oscillate backwards and forwards, 
and to return more rapidly than it advances. 

The length of the stroke made by Q must be regulated by the 
character of the work done, and is made greater or less by shifting 
R farther from or nearer to B. This adjustment does not affect the 

FIG. 114. 




inequality in the relation between the periods of advance and re- 
turn which the machine is intended to produce. (x-<^ ' 

ART. 89. As a further illustration of this slit-bar motion, we 
give a sketch of a curvilinear shaping machine used at the Crewe 
Locomotive Works. 

There have been instances of unequal wear of the tyres in the 
leading wheels of locomotive engines, which have been traced to 
the circumstance of the wheel itself being a little out of balance ; 
that is to say, the centre of gravity of the wheel did not exactly 
coincide with its centre of figure. 

In one case a wheel was found upon trial to be 9 Ibs. out of 
balance. 

Now we learn in mechanics that a weight of W Ibs. describing 
a circle of radius (r) with a velocity of (v) feet per second, will, 



I0 4 



Elements of Mechanism. 



during its whole motion, exert continually a pull upon the centre 
in the direction of a line joining the body and the centre, which 
will be measured in pounds by the expression 



32-2 x r 

Suppose a wheel 3 ft. 6 in. diameter to run at a velocity of 50 
miles an hour. In this case v will be equal to if?, and r will be -, 



and . _ = 

32-2 x r 



9 x 32-2 x I 
8 x 12100 
7 x 9 x i6'i 



whence the pull of only one pound weight at a distance of 3$ ft. 
from the centre will amount to rather more than 95 Ibs., and a 
weight of 9 Ibs. would produce a pressure upon the bearing of 
rather more than 75 cwts. ; and then, in the time of a half- revo- 
lution, viz., about y^th part of a second, the same pressure in the 
opposite direction. 

FIG. 115. 




It is, of course, only at high speeds that the defects due to 
want of balance become serious, and this numerical result shows 
very plainly the necessity of great care in the construction of wheels 
which are required to run at a high velocity. 



Mangle Wheels. 



105 



The machine intended to shape the curved inner face of the 
rim of locomotive wheels has the quick return movement which we 
have just discussed. 

The point B is the centre of motion of the lever bar, and 
coincides with the centre of the circuJar portion forming the inner 
surface of the rim of the wheel W. The tail end of the lever has 
a long slot in which the crank pin P works : this pin is attached 
to the driving disc centred at C, and the length of the stroke can 
be adjusted by shifting P in the direction of the radius CP. 

ART. go. Mangle wheels form a separate class of contriv- 
ances for the conversion of circular into reciprocating motion. 

A mangle wheel is usually a flat plate or disc furnished with 
pins projecting from its face ; these pins do not fill up an entire 
circle upon the wheel, but FIG. n6. 

an interval is left, as shown 
at F and E. 

A pinion, P, engages 
with the pins, and is sup- 
ported in such a manner as 
to allow of its shifting from 
the inside to the outside, 
or conversely, by running 
round the pins at the open- 
ings F and E. 

The pinion, P, always 
turns in the same direction, 
and the direction of rota- 
tion of the mangle wheel is the same as that of P when the pinion 
is inside the circular arc, and in the opposite direction when the 
pinion passes to the outside. 

The mangle wheel may be converted into a mangle rack by 
placing the pins or teeth in a 
straight line. Here the pinion 
must be so suspended as to 
allow of its shifting from the 
upper to the under side of the 
rack. 

As to the velocity ratio between the wheel and pinion, it will 




FIG 




io6 



Elements of Mechanism. 



be shown hereafter that the inner and outer pitch circles coincide 
in the case of a pin wheel, and therefore that the relative rotation 
of the mangle wheel to the pinion is the 
same in both directions. 

If the pins be replaced by a curved 
ring furnished with teeth, the mangle 
wheel will move more rapidly when 
the pinion is upon the inside circum- 
ference, and by giving certain arbitrary 
forms to this annulus, the velocities 
of advance and return may be modi- 
fied at pleasure. Contrivances such as 
this are seldom met with at the present 
time. 

ART. 91. Sometimes the pinion is fixed, and the rack shifts 
laterally. An excellent form of this arrangement was introduced by 
Mr. Cowper, and serves to give a reciprocating movement to the 
table in his printing machine. 

FIG. 119. 





The rack HF is attached to the system of bars in the manner 
exhibited in the diagram. A and C are centres of motion, and 
are the points where the bars are attached to the table. AG and 
CE are bisected in B and D, and are joined by the rod BD ; the 
rack HF is attached to the bars AG and CE by the connecting 



Mangle Motion. 107 

links GH and FE, and it must be remembered that it is the in- 
tention to obtain this so-called mangle motion by the reverse 
process of fixing the pinion and causing it to drive a continuous 
rack which runs upon each side of it alternately. 

The precise value of the contrivance consists in the arrange- 
ment of the bars, which will be understood upon referring to the 
section upoh Parallel Motion, and it will be seen that when the 
pinion has pushed the rack to either end of its path, the bars will 
so act 'as to move together, and will shift this rack HF to the 
opposite side of the pinion, without allowing it to deviate from a 
direction coincident with that in which the table is moving. 

This is the object of the contrivance, and, as we have said, the 
method by which the result is arrived at will be apparent when the 
subject of parallel motion has been examined. 

Thus the table carrying the parallel bars and the rack oscillates 
backwards and forwards, while the pinion, which transmits the 
force, remains fixed in space. 

When this machine was applied to the printing of newspapers, 
the table moved at the rate of 70 inches in a second, and its 
weight, including the form of type, would be about a ton and a 
half. When urged to its highest speed the machine would give 
5,500 impressions in an hour, which is about the greatest number 
attainable under a construction of this kind ; the true principle in 
rapid printing being that announced in the year 1790 by Mr. 
Nicholson, who proposed to place the type upon a cylinder having 
a continuous circular motion, and upon which another cylinder 
holding the paper should roll to obtain the impression. But 
although Mr. Nicholson enunciated the principle nearly a hun- 
dred years ago, and took out a patent for a mode of carrying 
it out, there is a wide difference between saying that a thing ought 
to be done, and showing the world how to do it in a practicable 
manner; hence it was not until late years that Mr. Applegath, 
and finally Mr. Hoe, were enabled so to arrange their cylinder 
printing machines upon the principle of continuous circular motion 
as to satisfy the wants of the daily papers, and to print some twelve 
or fourteen thousand sheets ir| an hour. 

To recur to our shifting rack, it must be remarked that by 
reason of the great weight of the table, and the rapidity with which 



io8 Elements of Mechanism. 

it moves, it would be quite unsafe to leave the rack and pinion in 
the present unassisted condition ; a guide roller therefore deter- 
mines the position of the pinion relatively to the rack, while the 
rack itself shifts laterally between guides. 

But since, theoretically, the rods would cause HF to move 
always in a direction parallel to itself, they practically enforce the 
desired movement in the path of the guides, with as little loss of 
power as possible. 

ART. 92. If it be required that the reciprocation shall be in- 
termittent, *>., that there shall be intervals of rest between each 
oscillation, we may employ a segmental wheel and a double rack, 
as shown in fig. 120. 

The teeth upon the pinion engage alternately with those upon 
either side of a sliding frame, and the motion is of the character 
required. The intervals of rest are equal, and are separated by 
equal periods of time. 

A pin upon the wheel and a guide upon the rack will ensure 
the due engagement of the teeth. 

FIG. 120. 





A mechanical equivalent to the above is found in the use of 
two segmental wheels and a single rack (fig. 121). 

These segments must be equal, but they may be placed in 
different relative positions upon the discs to which they are 
attached ; and, as a consequence, the intervals of rest may be 
separated by unequal periods of time. 

These segmental wheels have been employed in the earlier 
days of mechanism, and there was a well-known instance in Mr. 
Cowper's printing machine, where a segment of a wheel engaged 
with a small sector at each revolution, and so fed on the sheets of 
paper by the push given while the segments were in action. 



Segmental WJieets. 



109 



Sir J. Whitworth has proposed the 
for the reversal in a machine for cutting 
further example of the use of these 
wheels, which, however, should always 
be avoided if possible. There is only 
one driving pulley, and two segmental 
wheels are keyed upon the driving 
shaft. They are close together in the 
machine, and for the sake of the ex- 
planation we have placed one above 
the other. The object is to effect the 
reversal of a shaft C : the segmental 
wheels A and A' have teeth formed 
round one half of each circumference, 
and the toothed segments are in situa- 
tions opposite to each other, as in fig. 
122. 

When the action of A ceases, that of 
the wheels A and C, or the wheels A', 
action, i.e., we have a reciprocation of C. 
of the case given in Art. 75. 



subjoined arrangement 
screws : we take it as a 




A' begins, and we have 
B, and C alternately in 
This is a direct example 



HO Elements of Mechanism. 



CHAPTER III. 

ON LINKWORK. 

ART. 93. In the present treatise the term 'linkwork' is ap- 
plied to combinations of jointed bars movable in one plane, the 
joints being pins, whose axes are respectively perpendicular to the 
plane in which the bars move. The crank and connecting rod is 
an example of linkwork, but the present chapter deals principally 
with combinations of three or more links. 

As a fundamental case, and one which will repay the trouble 
of examining it, we take two cranks or levers centred at a 
distance from each other, mov- 
able in one plane, and connected 
at their extremities by a jointed 
bar. 

Such a combination is repre- 
sented by CPQB, where C and 
B are fixed centres of rotation 
f B formed by two parallel cylindric 

pairs, the arms CP, BQ being 

cranks or levers movable about axes through C and B, while PQ 
is technically known as a link or coupling rod, and is attached to 
the respective cranks by pins. 

There are, in fact, four parallel cylindric pairs, and three 
movable bars. 

Prop. When two unequal arms or cranks are connected by a 
link, as in fig. 123, the angular velocities of the arms are to each 
other inversely as the segments into which the link divides the line 
of centres. 

i. This may be proved by reference to an instantaneous axis. 
Upon causing the combination to rack a little, it will be found 




Two Cranks and a Link. in 

that P begins to describe a circle round C, while Q begins to 

describe a circle round B, whence it becomes evident that the 

instantaneous centre, F IG . , 24 . 

about which PQ is ro- 

tating at any instant, lies 

in the point O, where 

CP and BQ meet when 

produced. 

Produce PQ to meet 
BC in E, and draw CS, R J2/ 

OF, BR respectively per- 
pendicular to PQ. Let G 

PCE=0, QBC=<?>, and when the figure racks a little, let CP, BQ 
describe the small angles dd and d^. At the same time P and Q 
will each describe the same small angle n about the centre O. 




. CP ^ = BQ . 

^9_BQ OP_BR OF_BR_BE 
* V<2~OQ X CP~OF X CS~ CS~CE' 

2. The same may be proved by the resolution of velocities. 

Let P and Q shift to / and q during the smallest conceivable 
interval at the beginning of the motion. 

Then the resolved part of the motion of Q in the direction QP 
is ultimately equal 

to Q? cos ?QP, FlG " I25 ' 

=Q? sin BQR 




= BRx angle QB^. 

So also the resolved part of the motion of P in the direction 
QP=CS x angle PC/. 

But in the first instant of the motion these resolved parts are 
equal to each other, because PQ remains for a brief space parallel 
to its first position. 



112 



Elements of Mechanism. 



:. BR x angle QB?=CS x angle PC/, 
. angle QB^_ CS 
' * angle PC/~BR' 

But although the angular velocities of the arms BQ and CP 
change continuously, yet they will be at any instant in the same 
proportion as the limiting ratio of the angles described by these 
arms in a very minute interval of time, the relative motions of the 
arms not being supposed to change during that interval. 
TT angular vel. of CP_angle PC/ 
angular vel. of BQ angle QB^ 

_BR = B:E 
~cs CE' 

3. By analysis, the same may be proved. 
Let CP=al CS=/n PQ 



8 and as before, and the angle CEP being equal to //. 




d=. a cos (0-t d) + c cos ^-b cos (< -f \L) 
'. by differentiation we have 

o a sin (0 + ip) (dQ + d^) c sin ^ 

+b sin 
.'. csin 



But k /i = ^ 
/.o = 



Four-bar Motion. 113 

. angular vel. of CP_BR_BE 

' 'angular vel. of BQ~CS ~CE' 

which proves the proposition in its general form. 

ART. 94. Let it be required that one crank (viz. CP) shall 
sweep round in a circle, while the other (viz. BQ) oscillates to and 
fro through a given angle. 

In the diagram, the point P describes a circle, while the point 

FIG. 127. 




Q oscillates to and fro in a circular arc, the points H and E mark- 
ing its extreme positions. 

Then CH = PQ-CP, 
CE = PQ+CP. 

Also since CP makes complete revolutions, it is essential that 
CP and PQ should come into a straight line before BQ and PQ 
have the power to do so. Hence we must have 
CB + BQ greater than CE, 
and CB-BQ less than CH. 

It will be readily seen, upon testing this statement, that if CB 
be taken equal to PQ, the crank CP will revolve, and BQ will 
oscillate, so long as CP is sensibly less than BQ. 

The angle of swing of BQ increases also as CP becomes 
more nearly equal to BQ, and tends to reach two right angles as 
a limit. 

Ex. Let CP 2, CB = PQ= 15, BQ= 8, to prove that BQ 
will oscillate through 30 from a position perpendicular to CB. 

When BQ is at the end of its swing on the right hand we 
have, since CE=iy, 



114 Elements of Mechanism. 



2 x 15 x8 
.'. CBQ = 90, or BQ is vertical. 

When BQ is at the end of its swing on the left hand we have, 
since CH=i3, 

cos CBQ= 22 5 6 A-i6 9= i2o = i 

2 X 15 X 8 240 2 

.'. CBQ = 60, which proves the statement. 

^ ART. 95. It has been stated that an extreme case occurs 
when CP = BQ, and when the connecting link PQ is also equal 
to the distance between the centres, viz., CB. 

Under these circumstances one crank, as CP, will make com- 
plete revolutions while the other, viz , BQ, oscillates through 180. 
But at the same time CPBQ may form a parallelogram, whose 
opposite sides and angles are equal, and if any provision be made 
for retaining PQ parallel to CB, the crank BQ will no longer os- 
cillate but will perform complete revolutions. 

In the diagram, CP, BQ represent two cranks connected by 

FIG I2g a link PQ, which is equal to CB, then 

it is apparent that CPQB forms a paral- 

C B lelogram so long as PQ remains parallel 

N. \ to itself, or that the parallelism of PQ 

^ ^ is the condition which ensures the joint 

rotation of the cranks. 

It is/ also apparent that when the driving crank comes upon 
the line of centres the joint CPQ will bend if there be any re- 
sistance to the motion in the follower, and BQ will then return, 
while CP alone continues its rotation. 

This state of things is shown in fig. 1 29, where CP sweeps round 
in a circle, while BQ oscillates through 180. The difference is 
FIG. 129. that in the first case PQ remains 

parallel to itself, and that in the 
15 second case it oscillates through 

an angle 20, such that 



But the oscillation is put an end to by superposing a second 




Four-bar Motion. 



FIG. 130. 



V4 




combination exactly similar to the first, as in fig. 130, where the 
bell crank lever PCP' is connected 
with a second identical bell crank 
QBQ', by means of the equal links 
PQ, P'Q'. 

In this way, when PQ is passing 
through the dead points, P Q' will 
hold it in a parallel position, and 
each connecting rod will prevent 

the other from taking that oblique position which is destructive of 
the required motion. 

This is the principle of the coupling link between the two 
driving wheels in a locomotive engine. There are always two 
links, one on each side of the engine, and the cranks are of 
course at right angles. 

The necessity of the second pair of cranks, with their link, is 
obvious upon a little consideration, and may be made very clear 
by constructing a small model of the arrangement ; it is only 
necessary to make the links move in different planes so that they 
may be able to pass each other. 

ART. 96. We next observe that any point in PQ will describe 
a circle equal to either of the circles described by P or Q, so long 
as PQ remains parallel to itself, and hence that a third crank 
equal to either CP or BQ, and placed between them, would be 
driven by PQ, and would further prevent PQ from getting into an 
oblique position at the dead points, or would produce the same 
result as the second pair of cranks with the link in the locomotive 
engine. 

Again, the same would be 
true of any point R in a bar SR 
connected rigidly in any way 
with PQ, the point R would de- 
scribe a circle equal to either of 
the primary circles so long as PQ 
remained parallel to CB. 

Also, if a crank were supplied 
at ER, the three cranks would 
go round together, and PQ would remain parallel to itself. 

I 2 



FiG. 




Elements of Mechanism. 



FIG. 132. 




Conceive now that three equal cranks, P/, Q^, Rr, are cen- 
tred at equal distances along a circle PQR, as shown in fig. 132, 
and let a second circle pqr, equal to 
PQR, be jointed to the cranks at 
the points, /, q, r. 

If the circle pqr be shifted so 
fhatj the cranks are allowed to rotate, 
each of them will describe a circle, 
the respective cranks will always re- 
main parallel to each other, and the 
circle pqr will move in such a man- 
ner that any line drawn upon it re- 
mains always parallel to itself. 

Hence the circle pqr may be 
employed as a driver to rotate all 
three cranks at the same time, and 
while doing so, it will itself sweep round without the slightest 
movement of rotation upon its own centre. 

It has what is sometimes called a motion of circumduction. 
ART. 97. A very small alteration in the construction will 
give a combination which has been useful in rope-making ma- 
chinery, and which was the first movement suggested for feathering 
the floats of paddle-wheel steamers. 

Let the centres of the two circles PQR and pqr be made fixed 
centres of motion, and let P/, Q^, Rr, 
remain as before. 

A power of rotation will now be 
given to both the circles, and P/>, Q^, 
Rr will be the connecting links which 
always remain parallel to CB, the line 
of centres. That is to say, the rotation 
of the circle PQR, about the centre C, 
will cause an equal rotation in the 
circle pqr, about its centre B, and P/, 
Qff, Rr, must remain parallel to CB. 
It is the same combination that we 
started with, under a different aspect, 
by reason that the proportionate si^e of the pieces has been 



FIG, 133. 




Feathering Paddle Wheel. 117 

changed. The two circles have been enlarged and brought to- 
gether, so that their circumferences overlap, and CP, B/> are the 
parallel cranks. 

Regarding the contrivance as a method of feathering the 
floats in paddle wheels, we find that, in the year 1813, Mr. 
Buchanan patented a form of paddle wheel in which one circle, 
as PQR, carried the floats, and another circle, pqr, rotating with 
the former, held these floats always in a vertical position, and so 
made them enter the water edgeways, instead of striking it obliquely 
with the flat surface, as is the case in an ordinary paddle wheel. 

ART. 98. This wheel of Buchanan has not been used for 
very sufficient reasons. It is not a good arrangement for the floats 
to enter the water in an exactly vertical line, because the motion 
of the vessel must compound with that of the floats, and the sup- 

FIG. 134. 




posed vertical path will not be one in reality, any more than It 
would be in the case of a stone dropped from the same vessel 
into the water. The stone appears to fall in a vertical line, but is 
really projected forwards. 

According to this view the float should enter the water at an 
angle such that its line of direction will pass through the highest 



Elements of Mechanism. 



point of the wheel, this being the direction of the resultant of the 
two equal velocities impressed upon a point in the wheel, the one 
being that due to the vessel, the other being due to the rotation of 
the wheel. 

This result may follow very closely from the construction in 
the drawing, which represents Morgan's wheel, where the floats are 
connected by rods with a ring that rotates round a fixed centre in 
the paddle-box. The floats are attached to small cranks and 
pivoted upon centres, one of them (the lowest in the drawing) 
being driven by a rigid bar which springs from a solid ring. Each 
float passes the lowest point in a vertical position, and is some- 
what inclined when entering or leaving the water. 

ART. 99. In the conversion of circular into reciprocating 
motion by two cranks and a connecting link, it is a condition of 
the movement that the connecting link shall swing through a 
given angle. 

This fact has been usefully applied in wool-combing machinery, 
and in 1852 Messrs. Lister and Ambler patented (No. 13,950) an 
improved arrangement for transferring wool from one carrying 
comb to another in the act of working the same. 

FIG. 135. 





The diagram represents the combination CPBQ, the connect- 
ing link QP being prolonged to E, and carrying a comb at its 
extremity. H and F are two fixed combs, and the object in view 
is to detach a lock of wool from H by the comb E, and to transfer 
it to F. The proportions of the parts are so chosen that CP per- 



L em ie lie's Ventilator. 



119 



forms complete revolutions, and the positions of the passing comb 
are shown. In the left-hand figure E is in the act of rising 
through the wool at H, and detaching it, while the other sketch 
shows E as about to deposit the tuft on the distant comb F. 

ART. 100. The changes in the position of PQ may also be 
applied to a useful purpose when both the arms CP and BQ make 
complete revolutions. 

A remarkable instance occurs in a mechanical ventilating 
machine proposed some years ago by M. Lemielle as a substitute 
for furnace ventilation in mines. 

FIG. 136. 




The apparatus consists of a circular chamber of masonry com- 
municating by an air passage A with the shaft of the mine on one 
side, and having a discharge opening D on the other side. Within 
the chamber is placed a revolving drum centred on an axis at C, 
and having shutters or vanes, as PQ, connected by rods, as BQ, 
with a second fixed axis at B. 

It is apparent that CPBQ forms our well-known combination, 



I2O 



Elements of Mechanism. 




and, with the proportions in the drawing, it is also clear that CP, 
BQ will both make complete revolutions. 

In doing so, PQ opens out and closes up again, sweeping out 
before it the air in the open portion of the chamber, and driving 
the mass before it from A to D. 

The several positions of PQ are shown at pq y p'q' , and it will 
be seen that PQ begins to open out on the left-hand side, and gets 
to its extreme position at or near to pq. In practice there are at 
least three vanes employed, as shown. 

A ' Lemielle ventilator ' has been constructed and set to work 
where the chamber is 14 ft. in diameter and 7 ft. deep, with a fan 
making some 37 revolutions per minute, and discharging in that 

,e 25,000 cubic feet of air. 

ART. 101. In sewing machines the 'shuttle race' or path of 
the shuttle is commonly a straight line, but the shuttle is some- 
times caused to oscillate to and fro in a circular arc. 

The drawing shows an arrangement for this purpose, founded 
upon the above combination of two cranks and a connecting link. 

Here the shuttle S is carried in a frame which oscillates on a 
fixed centre at B. The driver is a pin P placed on the face of a 
cam-plate whose centre is at C, and connected by a link PQ to 
the stud Q attached to the frame carrying the shuttle. The plate 
carrying P is not circular, and its edge A is used as a cam driver 
for another part of the operation of the machine. 

FIG. 137. 




It is apparent that we have assigned such dimensions to the 
working parts, viz., CP, BQ, and PQ, as will produce the result 
that Q will oscillate while P performs complete revolutions. 



Hand Shearing Machine. 121 

Accordingly, the left-hand diagram shows two extreme posi- 
tions of S and of the frame. It will be seen that P describes a 
circle of radius CP while Q moves through a portion of the dotted 
circular arc formed round the centre, B. The carrying backwards 
and forwards of the shuttle is thus arrived at very simply. ^. 

ART. 102. The combination CP, BQ, and PQ appears in th 
form of a hand machine for cutting metals. 

The diagram will make the construction quite apparent. The 
lever BH, having a handle at H and a fulcrum at B, is connected 
by the link PQ with a second piece CP carrying a knife-blade. 
This latter piece has a centre at C, and the cutting is performed 
by depressing the handle H. 

FIG. 138. 




As a question of mechanism it will be found that the angular 
motion of CP is much less in amount than that of BH, and the 
principle of work comes in and affords an easy explanation of the 
power of the machine. 

A skeleton diagram will show the mechanical advantage, 
Let S be the force applied at H, 

R the resistance in cutting the metal as felt at D. 
Then, if T be the thrust in QP, we have 
TxBQ = S xBH, 
RxCD = TxCP. 
/. R x BQ x CD = S x BH x CP 



122 Elements of Mechanism. 

Ex. Let CD = BQ = i, CP = 4, BH = 10, 

ThenR = Sx -^4 
i x i 

= 408, 

which is a much better result than would be obtained by the use 
of a single lever CH carrying a knife. 

ART. 103. The combination of two unequal arms with a con- 
necting link forms the celebrated Stanhope levers which have been 
so generally employed in printing presses worked by hand. 

From the invention of the Art of Printing in the year 1450 till 
the year 1798 no material improvement was made in the printing 
press. The earliest representation of a press occurs as a device in \ 
books printed by Ascensius. There is scarcely any difference be^ V 
tween it and a modern press, and it is truly a matter of astonish- v 
ment that so long a period as 350 years should have rolled on\\ 
without some improvement being made in so important a machine, x 

The wooden press consists of two upright pieces of timber > 
joined by transverse pieces at the top and near the bottom ; a 
screw furnished with a lever works into the top piece, and by its 
descent forces down a block of mahogany, called the 'platten,'and 
thus presses the sheet of paper upon the type, which is laid upon 
a smooth slab of stone embedded in a box underneath. In the 
year 1798 Lord Stanhope constructed the press of iron instead of 
wood, and at once transferred the machine from the hands of the 
carpenter to those of the engineer ; he further added a beautiful 
combination of levers for giving motion to the screw, causing 
thereby the platten to descend with decreasing rapidity and conse- 
quently increasing force, until it reached the type, when a very 
great power was obtained. 

These levers consist of a combination of two arms or cranks, 
CP, BQ, connected by a link, PQ, in such a manner that the con- 
necting link shall come into a position perpendicular to one of the 
arms at the instant that it is passing over the centre of motion of 
the other arm. 

In order that this may happen, it is evident that the various 
pieces must satisfy the relation. 



PQ_CP= N /CB 2 -BQ 2 . 



The Stan hope Levers. 



123 




For the convenience of the workman who is employed upon 
the press, a handle, CA, is attached to the crank, CP, and moves 
as part of it, but the intro- FlG I3g 

duction of this handle does 
not affect the principle of 
the movement, which, re- 
garded as a question in 
mechanics, depends simply 
on the combination of CP, 
BQ, and PQ. 

If, now, a force, F, be 
applied at the end of the 
handle, AC, so as to turn 
the crank, CP, uniformly in 
the direction indicated, the 
arm, BQ, will, under the 
conditions already stated, 
move with a continually 

decreasing velocity until it comes to rest, and then any further 
motion of CP will cause BQ to return. 

The lower diagram shows the levers in this extreme position, 
and the graduated scales at P and Q indicate the relative angular 
movements of CP and BQ. 

Now the motion, interpreted with relation to the transmission 
of force, implies that the resistance at Q necessary to balance the 
moving power which turns the crank is increasing rapidly as the 
rotation of BQ decreases, and that there is no limit theoretically 
to the pressure which will be felt as a pull at Q by reason of the 
force F. In practice this extreme pressure is exerted through so 
very small a space that the theoretical advantages are scarcely 
realised, but the arrangement is exceedingly useful as applied in 
the printing press. 

The lever BQ is there employed to turn the screw which acts 
upon the platten ; tne workman gives a pull to the handle AC, 
and by doing so causes the platten to descend with a motion which 
is at first considerable, and afterwards rapidly dies away. Thus the 
limited amount of power which is being exerted comes out with 
greatly magnified effect in impressing the paper upon the type. 



a, ^ 

124 Elements of Mechanism. * ** 



ART. 104. In order that this contrivance may be better under- 
FIG. 140. stood, take the annexed sketch to 

c B represent it, and draw CN perpen- 

S[\ \ dicular to PQ. 

J^ \ \ \ A force F acting at A in a direc- 

s N \ s o tion perpendicular to CA would be 
balanced by a force S acting in PQ, 
such that 

S x CN = F x CA, 
o FxCA 

vS : ~~CN~' 

' This force S, necessary to balance F, would be supplied by the 
resistance to motion in the arm BQ, and would, in fact, be the 
pull felt at Q. Now, as the arms turn, the link PQ gets nearer and 
nearer to C, and CN becomes less and less until it has no appre- 
ciable magnitude, and the consequence is that S increases enor- 
mously in the last instant of the motion. 

Ex. Let F = 20 Ibs., CA 20 inches, CN = T Vth of an inch, 
we have 




ART. 105. In the Stanhope levers the work is completed just 
as PQ overlaps CP, and there is no advantage to be gained in 
carrying the motion further ; but regarding the combination in its 
general form, it has been shown that the joint CPQ straightens 
into a line twice while CP is making a complete revolution. 

When this happens we obtain a subordinate combination of 
levers, which is known as a knuckle joint, or toggle joint, and is 
commonly used in hand printing presses, as well as in machinery 
for punching and shearing iron. 

The first form of the joint is shown in Fig. 141, where two 
arms, CP, PQ, generally equal 
in length, but which may be 
unequal, are jointed together, 
the point C is fixed, and the 
end Q exerts a pressure which 
may be carried on by a piece 
moving in the direction CQ. 




Toggle Joint. 125 

The force F, which produces the result, is supposed to act upon 
the joint at P, and the reaction S, which is felt at Q, will be trans- 
mitted also to P in the direction QP. 

Let the thrust, so set up in QP, be called R, and draw CN 
perpendicular to PR. 

Then, by the principle of the lever, we have 

R x CN = F x (perpendicular from C upon F). 

But as CPQ straightens, CN diminishes, and ultimately becomes 
equal to zero while the product Fx (perpendicular from C upon 
F) remains approximately constant. It follows, therefore, that the 
product of R x CN remains nearly constant while CN diminishes 
to nothing, but this can\pnly happen if R increases in the same 
proportion that CN diminishes, that is, without limit. Hence the 
power of the combination. 

It is upon this principle that the heavy chain of a suspension 
bridge cannot be stretched into a straight line ; it would break long 
before it straightened. 

In the same way, if the joint be doubled back, the point Q 
being fixed, and the force F being supposed to act in a line per- 
pendicular to CQ, we shall 
have the pull upon C, which 
we may call R, felt as a pres- 
sure in the line CP. 

If now QN be drawn per- 
pendicular to the direction of 
R, as felt at P, the principle of the lever gives us the equation 

RxQN = FxQM, 

and R becomes infinite when QN vanishes, or when CP is passing 
over Q. 

ART. 1 06. If the toggle joint be regarded merely as a me'ans 
of communicating a slow motion to the end of one of its pair of 
levers, it is not without practical utility. 

But inasmuch as any force which may be transmitted through 
such a joint must be intensified at the period of slow motion, we 
shall generally find that some work is being done which demands 
an increase of pressure. 

The joint is often attached to a revolving crank and connecting 
tod as shown in the diagram, where we have the combination 




126 Elements of Mechanism. 

CPQB as in Art. 105, with the addition of a rod EQ jointed to Q, 
and guided so that the end E moves in the line BR. 

FlG . I43 . In this case C should be in the ver- 

tical line passing through Q when the 
joint is straightened, and PQ should be 
just long enough to reach the lowest 
point of the circle at the same instant. 
It will follow that the joint can, under 
these circumstances, straighten itself 
only once during a complete revolution 
of CP, and the contrivance may then 
be applied so as to obtain a decreasing 
motion of the point E, and thereby to 
transmit a pressure which greatly and 
rapidly increases. 

An example occurs in printing machinery, where a knuckle 
joint, actuated by a crank exactly as described above, is employed 
to depress the platten upon the impression table, and so to effect, 
in a large machine worked by steam power, the same thing which 
is done on a smaller scale by the pull of the lever of a hand press. 
In the case last treated, the point Q never passed below the line 
BR, and thus the joint only straightened itself once during a revo- 
lution of CP ; it is possible, however, to cause this straightening 
to occur twice in each revolution of the crank, and to effect this, it 
is only necessary to shift the point C a little nearer to BR, so that 
the joint may straighten when P is upon either side of the lowest 
or highest point of its circular path. 

The sketch shows the knuckle joint as applied to a movement 
of this character in a power loom (fig. 144). 

In this case the joint will straighten when P arrives at certain 
points on either side of the vertical line CA, as shown by the posi- 
tions of CP, CP', and thus we shall find that the point Q falls below 
BE, as well as rises above it, and that there will be two positions 
of P upon either side of A in which BQE becomes a straight line. 
In weaving, the thread of the weft requires to be beaten up 
into its place after each throw of the shuttle ; and in some cases, 
as in carpet weaving, two beats are wanted instead of one. 

The arrangement which we are now discussing has been used 



Multiplied Oscillation. 



127 



to actuate the movable swinging frame, or batten, which beats up 
the weft, and the result is that two blows are given in rapid suc- 
cession. 

FIG. 144. 




In the figure referred to, EF is the batten, movable about F 
as a centre, and it is clear that when the crank takes the positions 
CP, CP', the joint BQE will straighten, and, as a consequence, the 
batten will be pushed as far as it can go to the left hand, or a beat- 
up of the weft will take place. 

We thus solve the problem of causing a reciprocating piece to 
make two oscillations for each complete revolution of an arm with 
which it is connected. 

ART. 107. This principle of obtaining two vibrations of a 
bar for each revolution of the driving-crank may be extended still 
further, and we will alter the construction so as to obtain four 
vibrations instead of two. 

The arrangement of the knuckle joint BQE and the crank CP 
remains as before, but the arm QE is now connected with a second 
knuckle joint FDL, and the piece AK, which is to receive four 



128 



Elements of Mechanism. 



vibrations, is centred at A, and is attached at the point L to the 
linkDL. 

FIG. 145. 




It is clear that the joint FDL will straighten itself four times in 
each revolution, viz., when the crank CP is in the positions marked 
i, 2, 3, 4, upon the circle, and thus AK will make four complete 
vibrations for each revolution of the crank. 

In other words there are two positions of the joint BQE in 
which FDL straightens, and each of these positions of BQE is 
obtained by two distinct positions of CP. 

By recurring to the earlier part of the chapter, we shall under- 
FlG J46 stand that a cam -plate movable 

about C, and shaped as in Fig. 146, 
may be employed to drive the bat- 
ten, and may replace the above com- 
bination, being, in point of fact, a 
mechanical equivalent for it. 

The roller P is then connected 
with levers attached to the batten, 
and the beat-up occurs when P 
passes through the hollows upon 
each side of the projection at C. 




Vatt's Parallel Motion. 



129 



ART. \c$r The \Parallel Motion used in steam-engines was 
the invention of Janies Watt, and was thus described by himself 
in the specification of a patent granted in the year 1784 : 

' My second new improvement on the steam-engine consists 
in methods of directing the piston rods, the pump rods, and other 
parts of these engines, so as to move in perpendicular or other 
straight or right lines, without using the great chains and arches 
commonly fixed to the working beams of the engine for that pur- 
pose, and so as to enable the engine to act on the working beams 
or great levers both by pushing and by drawing, or both, in the 
ascent and descent of their pistons. I execute this on three prin- 
ciples. . . . The third principle, on which I derive a per- 
pendicular or right-lined motion from a circular or angular motion, 
consists in forming certain combinations of levers moving upon 
centres, wherein the deviations from straight lines of the moving 
end of some of these levers are compensated by similar deviations, 
but in opposite directions, of one end of other levers.' 

The annexed sketch is copied from the original drawing depo- 
sited in the Patent Office. 



FIG. 147. 




LI 


J 


1 


T iJ 



AB is the working beam of the engine^ PQ the piston rod or 
pump rod attached at P to the rod BD, which connects AB and 
another bar, CD, movable about a centre at C. 

' When the working beam is put in motion the point B de- 
scribes an arc on the centre A, and the point D describes an arc on 
the centre C, and the convexities of these arcs, lying in opposite 
directions, compensate for each other's variation from a straight 
K 



1 30 Elements of Mechanism. 

line, so that the point P, at the top of the piston rod, or pump 
rod, which lies between these convexities, ascends and descends 
in a perpendicular or straight line.' 

ART. ^o^-This invention being an example of our combi- 
nation of two cranks and a connecting link, we proceed to discuss 
it in a careful manner, and to examine its peculiar features. 

The lines AB and CD in the diagram represent two rods 
movable about centres at A and C, and connected by a link, BD. 
If BD be moved into every position which it can assume, the path 
of any point P in BD will be a sort of figure of eight, of which the 
portions which cross each other are nearly straight lines. 

At the beginning of the motion let the rods be so placed that 
the angles at B and D shall be right angles. 

FIG. 148. FIG. 149 





We shall now endeavour to discover that point in BD which 
most nearly describes a straight line, and in doing so, we first re- 
mark that BD begins to shift in the direction of its length, and 
therefore that the straight line in question must coincide with BD. 

The exact position of the so-called parallel point, that is, the 
point P in Watt's diagram, is determined very simply by analysis, 
and we shall give the investigation immediately. But we can 
readily predict where it must be found. 

As stated by Watt, the points B and D describe circular arcs 
about the centres A and C, the convexities of these arcs lying in 
opposite directions, and if AB and CD be equal, the parallel point 
P must be so placed that its tendency to describe a curve with a 
convexity approaching to that of the path of B is exactly neu- 
tralised by its tendency to describe another curve with a like con- 
vexity in the opposite direction due to its connection with CD. 



Watt's Parallel Motion. \ 3 1 

Hence P must lie in the middle of BD, and being solicited by 
two equal and opposite tendencies, it will follow the intermediate 
course, which is a straight line. If, however, AB and CD are un- 
equal, the path of the point P will be affected by the increased 
convexity due to its connection with the shorter arm CD, and in 
order to escape from this effect it will be necessary to move P 
away from D, and to bring it nearer to the arm AB, whose ex- 
tremity traces out a curve of less convexity. 

It may be expected, since we are dealing with circular arcs, 
that the point P should now approach B in a proportion identical 
with that given by comparing AB with CD, or that we should have 
BP CD 



It is very easy to construct a small model, and to verify in this 
way the principle of Watt's parallel motion. 

If the arms AB, CD, be equal, but the describing point P does 
not bisect BD, but is brought near to the end D, as in fig. 149, the 
regular looped curve will become distorted and will incline towards 
C, in the manner shown in the diagram. 

ART. no. Refer now to fig. 150, and suppose the rods to be 
moved from the position ABDC into another position A.l>dC. 



FIG. 150. 

\ 



FIG. 151. 




Draw bm, dn perpendicular to AB and CD respectively, and let P 
be the point whose position is to be determined. 
Let AB = r, b? = x, BA = 6, 



132 Elernen ts of Median ism. 

We shall suppose in what follows that the motion of AB and 
CD is restricted within narrow limits, and shall deal approximately 
with our equations, by putting 

, rf> 
sin - = -, and sin - =-, 

22 22 

., x f>P Em 
the V == ^ = D 

_r (i cos0) 
s (i cosp) 

2 sin 2 ? 



s 2 sin 2 ?? 

2 

= s nearly. 
JY/ 

But the link only turns through a very small angle, which may 
be considered to be nothing as a first approximation, in which case 
the vertical motion of B is equal to that of D, 

.-. bm dn, or r sin = 5 sin 0, 
whence r 6 = s <t> nearly. 

* * y s r 2 /' 
P _ CD 

r Pd AB ' 

/>., the point P divides BD into two parts which are inversely as 
the lengths of the nearest radius rods. 

In the case considered, which is that which occurs in practice, 
the parallel point lies in the connecting link, but if the rods be 
arranged on the same side of the link, as shown in fig. 151, the 
required point will lie in BD produced, and on the side of the 
longer rod. 

Suppose now the rods to be moved into the position AbdC, 
and draw bp, dl perpendicular to BD and BD produced re- 
spectively. 

' _ bp _ r (i-cos 0) _ r 2 
*~ dl~ s (i cos y) ~ 7y* 



Watt's Parallel Motion. 
Also r =stp , by parity of reasoning, 
'^P = ^ X ^ = ? 



133 



and the point P obeys the same general law whether it be found 
in the link itself or in the prolongation of the line of its direction. 

ART. 1 1 1. We have supposed that sin Q = Q and sin <p = $ in 
the previous investigation, and have examined the motion of that 
point in the connecting link which most nearly describes a straight 
line. We shall now inquire how much P really deviates from the 
rectilinear path at any given period of its motion. 

In practice, the beam of an engine seldom swings through an 
angle of more than 20 on each side of the horizontal line, and 
within that limit the error consequent upon our assumption that 
the sine of an angle is equal to its circular measure would not be 
considerable : for we find, upon referring to the tables, that the 
circular measures of angles of i, 5, 10, 15, 20 degrees, and the 
natural sines of the same angles are the following : 



Angle 


Circular Meas. 


Natural Sine 


Difference 


1 


0174533 


0174524 


0000009 


5 


0872665 


0871557 


ooonoS 


10 


I74S329 


1736482 


0008847 


15 


2617994 


2588190 


0029804 


20 


3490659 


3420201 


0070458 



In an engine of the usual construction AB is equal to CD, and 
we shall simplify our results by making this supposition. 

Let BD move into the position fid, and turn through an angle 
a ; it is very apparent that within the limits to which the rods 
move in practice, will be much less than 6 or ^, so that we may 
regard a as a close approximation to the actual value of sin a even 
when we do not adopt the same supposition with regard to 6 and 
^. The object of the investigation will be to determine <j> in terms 




1 3 }. Elements of Mechanism. 

of 0, and we shall see that the deviation sought for depends upon 

the difference of the cosines of </> and 6. 
FIG. 152. As before, observing that s = r, we have 

J$m = r (i cos 0), 
m b = r sin 0, 
d n = r (i cos </>), 
Dn = r sin (j>. 
Let BD = /, then / + mb = I cos a + ~Dn, 

or / + r sin 6 = I cos o + r sin </;, 
/. r sin (/> = ?- sin d + / (i cos a), (i). 

Now a being the angle through which BD is 
twisted, and being moreover very small, we shall have 
/a = B/ t dn^ very nearly, 
= r (i cos 0) + r (i cos 0) 
= 2 r ( i cos 0), since <f> is nearly equal to 9. 

/. a = 2 r (i cos 0) very approximately. 

By substituting in equation (i) we can calculate <j> with con- 
siderable accuracy, and then the deviation of P from the vertical 



and can therefore be ascertained. 

Ex. Let 6 = -, and assume r s = 50 in., / = 30 in. 
9 

.'. n = --' (-0603074) = : - (-0603074) = 2010247, 

or represents the angle 11 31'. 

Substituting in equation (i) we have 

sin <!>= sin 2o + 2(r cos 11 31') 

= -3420201 + 3 ('0201333) 

= -3541001. 

.*. <f> represents an angle of 20 44' nearly. 
Kence the deviation of P from the vertical 

= 5 (COS 20 -COS 20 44) = 25 (-0044544) 

= T ^th of an inch approximately. 



Similar Curves. 135 

It may be shown that this amount of deviation is again capable 
of reduction if we cause the centres of motion, A and C, to 
approach each other by shifting them horizontally through small 
spaces. 

ART. 112. The point B, whose motion has been examined, 
is usually found at the end of the air-pump rod. We have now to 
obtain a- second point, also describing a straight line, and suitable 
for attachment to the end of the piston rod. 

We require, in the first instance, to know when two curves are 
similar, and in a Cambridge treatise on Newton's ' Principia ' the 
test of similarity is stated in the following terms : 

Two curves are said to be similar when there can be drawn in 
them two distances from two points 
similarly situated, such that, if any 
two other distances be drawn equally 
inclined to the former, the four are 
proportional. 

Ex. Thus all parabolas are similar 
curves, and all ellipses with the same A - 
eccentricity are similar curves. 

Let A, A', be the vertices, S, S', the foci of two parabolas. 

Then SA, S'A', are two lines drawn from two points similarly 
situated, viz., the foci of the curves. 

Let SP, S'P' be radii inclined at the same L V to SA, S'A' 
respectively. 

ThenSP= 2SA -, 
i+cos 0' 




. 

i-f cos 

SP = SA 
' ' S'P' S'A" 

whence the curves are similar, and there is no exception to this 
rule. 

Those who are conversant with the properties of a parabola 
know very well that it represents, with great exactness, the path of 
a stone thrown obliquely into the air, and gives the theoretical 
form of the path of a projectile when unaffected by the resistance 
of the air. 



136 Elements of Mechanism. 

The similarity of all such curves to one another is by no means 
evident upon cursory observation, but it is at once established by 
this simple reasoning. 

In the case of ellipses, we proceed in a similar manner, and 
now S and S' represent the foci of two ellipses of eccentricity e 
and e 1 respectively. 



i + e cos 
S'A' 



. s /p/ 

i+e' cos 6 ' 
Let now e d, the eccentricities being identical, 



^ or the curves are similar only under the condition stated. 
*f ART. ^y. Without any further enquiry into the nature of 
A the curves which satisfy the condition of similarity, we will pass 
on to examine an extremely useful instrument called a Pantograph^ 
which is formed as a jointed parallelogram with two adjacent sides 
prolonged to convenient lengths, and is used to enlarge or reduce 
drawings according to scale. 

This parallelogram was incorporated by Watt into the inven- Kr 
tion of the parallel motion, and gave it that completeness which ity 
has at the present time. We have now to show that the pant^ 
graph is an apparatus for tracing out similar curves. y 

In the diagram, let BQRC represent a parallelogram whose 
sides are jointed at all the angles, 
and having the two adjacent sides 
BC, BQ, lengthened as shown. 
Take a point S, somewhere in BC 
produced, as a centre of motion, 
place a pencil at any point P' in 
the side RC, produce SP' to meet 
BQj or its prolongation in P, 
place another pencil at P, when 
it will be found that by moving 
about the frame over a sheet of 
paper, and at the same time 
allowing the joints free play, it will be possible to describe any 





The Pan tog rap/i. 1 3 7 

two curves that we please, and these curves will be similar to each 
other. 

Our definition tells us that the two pencils will trace out 
similar curves if we can show that SP always bears to SP' the 
same ratio that two other fixed lines radiating from S, and to 
which SP and SP' are equally inclined, also bear to each other. 

Conceive, now, that SCB originally occupied the position 
SA'A, and draw the line SA'A as a fixed line upon the paper, then 
we shall always have 

gp>= s ^=g^7, which is a constant ratio, 

and the angle ASP is equal to the angle A'SP', therefore the points 
P and P'will trace out similar curves so long as SPT remains a 
straight line. 

This is, therefore, the only condition which we have to observe 
in using the instrument. - 

ART. 114. We are now in a position to complete the Parallel 
Motion of a Seam Engine, for if one of the points describe a 
straight line, the other must do the same. 

To the system, ABCD, we superadd the parallelogram BFED; 
ABF being the working beam of the engine. 

FIG. 155. 



The usual construction is to make the arms AB and CD equal 
to each other, in which case P, which is the point to which the 
air-pump rod is fixed, will be in the centre of BD ; whereas the 
second point E, which we are about to find, lies in AP produced, 
and is the point of attachment of the end of the piston rod. 

In order to find the side BF in this parallelogram BFED, 
assume that AB = r, CD = j, BF = x. 

Then -= by similar triangles ABP, DPE. 
T PB 



1 38 Elements of Mechanism. 

p\T) A T) ^ 

Also -=---= -, by property of the parallel motion, 
PD CD s 

x r r 1 

.'. -=-, or x= , 

r s s 

which equation determines the proportion between BF, AB, and 
CD, in order that the second point sought for may lie at the 
vertex, E, of the parallelogram BFED. 

Thus we see that the complete arrangement consists of two 
distinct portions incorporated together. 

1. The combination of AB, CD, and BD, which compels some 
point P to describe a straight line> the position of this point de- 
pending upon the relation between AB and CD. 

2. The Pantograph ABFED, which has some point in FE, 
here for simplicity selected at E, that must necessarily describe a 
straight line parallel to the path of P. 

And, as we have stated, the points of attachment of the ends 
of the air-pump and piston rods to the main beam of the engine 
are thus provided for. 

ART. 115. Where a beam engine is used in a steam vessel 
the beam must be kept as low down as possible, and the motion is 
altered as in the figure, but it is precisely the same in principle. 

Beginning with ABDC, a system of two arms and a connecting 
link, we obtain the parallel point P ; we then construct the panto- 
graph CDBFE so as to arrive at the point P', whose path is 
similar to that of P. 

Here CE represents the beam of the engine, and P' is the point 
to which the end of the piston rod is attached. Draw GH parallel 

FlG . I56 . to FE, and let Z ^ BD = I FF= *' 

HD = GB = BP 

BP CD s 



.,HD =1 D= , 2 



s r r 

FFFFEH 



A Parallel Motion. 



139 




r+s 



whence thej^osition of the point P' is ascertained. 

AR^J^/ A Parallel Motion may also be useful in machinery. 

In the old process of multiplying engraved steel plates at the 
Bank of England, which was practised before the art of electro- 
typing was understood, it was necessary to roll a hardened steel 
roller upon a flat plate of soft steel with a very heavy pressure, and 
so to engrave the plate. The difficulty of maintaining this pressure 
during the motion of the roller upon the surface was overcome by 
the aid of the parallel motion shown in the drawing. 



FIG. 157. 




The system of jointed bars allowed the heavy frame C to 
traverse laterally, while the necessary pressure was obtained by a 
pull upon the end B of the lever AB, which lever was movable 
round A as a centre of motion, and was further connected at B 
with somersource of power. 

ART. (lira A straight line motion which is founded on the 
propositiorr in Euclid that the angle in a semicircle is a right 



140 



Elements of Mechanism. 



FIG. 158. 




angle has been suggested by Mr. Scott Russell. It is derived from 
the ordinary crank and connecting rod. 

Let the rod RQ be bisected in P, and jointed at that point to 
another rod CP, which is equal in length to PQ. Suppose the 
point C to be fixed as a centre of 
motion, and the end Q to be con- 
strained to move to and fro in the 
line CQ, then R will move up and 
down in a straight line pointing also 
toC. 

Since CP = PQ = RP, the point 
P will be the centre of a circle passing 
through C, Q, and R. 

Also RPQ is a straight line, and 

must therefore be the diameter of the circle, whence the angle ^ 
RCQ is a right angle, or the point R must always be situated in a /\r 
straight line through C perpendicular to CQ. 

That is, the path of R is a fixed straight line pointing to C. 
The motion fails after CP has rotated through two right angles \ 
from the upper vertical position. The point Q then gets back to 
C and remains there. 

The motion is rather one for copying a straight line than for 
generating it, for the truth of the straight line described by R will 
be neither greater nor less than that of the guide CQ which directs 
its motion. 

ART. 1 1 8. The movement maybe analysed on the doctrine 
of harmonics. Taking the rods in the position shown, it is appar- 



FIG. 159. 



ent that CP has moved from C/ and 
that P has completed a harmonic mo- 
tion ;//P in a horizontal direction towards 
the right hand. At the same time PR 
has turned round P towards the left 
hand through a circular arc rR. which 
is exactly the same as the corresponding 
path of P. The harmonic motion of R 
is therefore #R in a horizontal direction 
towards the left hand. 
But R receives the motion of P in addition to its own proper 




Straight Line Motion. 



141 



movement round P as a centre, and the horizontal components of 
these movements are equal and opposite. Hence R describes a 
vertical straight line. 

Hereafter it will be shown that the backward rotation of RP at 
the same rate as the forward rotation of CP may be provided for 
by wheelwork, and that a straight line motion may be obtained 
without any guide along CQ. 

ART. fj^-Since the path of P is a circle round C it follows 
that if AB7T>F, represent grooves on a plane surface, and the rod 
RQ has pins at R and Q FlG l6o- 

working in the grooves, the D 

circular motion of CP will 
cause R and Q to oscillate 
to and fro in AB and DF 
through spaces equal to 
4CP. And the converse is 
also true. B 

This suggests an in- 
structive model. 

For if CP and PQ repre- 
sent an ordinary crank or 
connecting rod, CP being 
equal \.o PQ, it will be found 

that the rotation of CP brings Q up to C and then the motion 
fails. The rods CP and PQ merely continue to maintain a circular 
motion round C. 

Whereas if QP be produced to R, such that RP = PQ, and a 
pin at R works in grooves lying in DF but not continued quite 
down to the centre C, the rotation of CP will cause Q to move 
from a distance 2CP on the right, up to C, and then to pass to a 
distance 2CP on the left of C, thus having a throw equal to four 
times the length of the crank. 

ART^5MP- Another form of parallel motion was devised for 
marine enfiries before the principle of direct action was so gene- 
rally adopted. It was fitted to the engines of the ' Gorgon ' by 
Mr. Seaward, and has since been applied in a modified form to 
small stationary engines, which are convenient in the workshop, 
and are known as Grasshopper engines ; but except so far as the 



I 4 2 



Elements of Mechanism. 



latter application is concerned, it has not been regarded with par- 
ticular favour. 

It is, however, remarkable as illustrating a mechanical principle 
for reducing the friction upon an axis, by causing the driving pres- 
sure and the resistance to be overcome to act upon the same side 
of the centre of motion ; for here the connecting and piston rods 
are both attached to the rocking beam upon the same side of its 
axis. In this respect it has an advantage, for in ordinary beam 
engines the pressure upon the fulcrum of the beam is the sum of 
the power and the resistance, whereas here it is the difference of 
these forces, and the friction is proportionally diminished. 

It is derived from Scott Russell's motion by replacing the guide 
by a portion of a circular arc of comparatively large radius. For 
the purpose of explanation we refer to the diagram, where the line 
HSP corresponds to the line RPQ in the last article, the point P 
moving very approximately in a horizontal line by reason of its 
connection with PQ, which has a centre of motion at Q, and the 
point H being that which most nearly describes a straight line. 
The system of rods is then TS, HSP, PQ, the points T and Q 
being centres of motion. 

Draw SR and HK perpendicular to TP, and SV perpendicular 
toHK, 

-.a\ cp _ STR=0 

:b } * - r 'SPR=$> 
FIG. 161. 



and let 




Then TR = acos0 = a ("1 2 sin 2 -^ 

(O 2 \ 
i J nearly, 



Peaucellur's Invention. 



cos <p b fi^ nearly, 



. TT<r , a q 

. . 1 K = a o --- 1 . 

2 2 

But the point H describes the straight line HK, 

/.TK = a-, 

M 2 aH 2 
whence we have -* ------ = o, 

2 2 

or aB* = b<f. 
But *=^ nearly, 



or SP is a mean proportional between TS and SH, a condition to 
be fulfilled by the rods giving the parallel motion. 

In a grasshopper engine PH is the working beam, and PQ is a 
vibrating pillar at one end, the piston rod is attached at H, and the 
connecting rod is jointed to some convenient point in the beam. 
y^ART. 121. The important discovery of the method of drawing 
' /li straight line by a combination of bars jointed together, and some 
of which are movable upon fixed centres, was first made public 
in the year 1864 by M. Peaucellier, an officer of Engineers in the 
French army. 

The combination consists of seven bars, as in Fig. 162, whereof 
CR = CS, C being a fixed centre of motion ; ED = DC, the 




point E being a fixed centre of motion ; and FR = RD = DS = SF, 
the whole of the respective bars being so jointed at their ends as 
to permit perfect freedom of motion in the plane of the paper. 
If the system be moved within the limits possible by its con- 



144 



Elements of Mechanism. 



struction the diamond-shaped figure FRDS will open out or close 
up, the points R and S will describe circles about C, and the point 
D will describe a circle, which, if completed, would pass through 
the point C, but F will describe an exact straight line. 

To prove this we refer to Fig. 163, where the bars have been 
moved upwards, one half of the combination being left out for 
greater simplicity in the diagram. 

FIG. 163. 




Join CP and produce it to Q, and draw RN perpendicular 
toPQ. 

Let CR = c, PR = 6, PN = NQ - x, NR =y. 



= CP(CP+2*) 

= CP x CQ. 

When the rods are in the normal position, as shown in fig. 162, 
let Q be at the point F. Join FQ and PD. 
Then by parity of reasoning we have 

CD x CF = ^-^= CP x CQ. 

.-. CD : CP::CQ : CF. 

Hence in the triangles CQF, CDP there is one angle common 
to each, viz., the angle QCF, and the sides about this common angle 
are proportional : therefore the triangles are equiangular and the 
angle CPD = angle QFC (Euclid, book vi. prop. 6). 

But CPD is the angle in a semicircle, and is therefore a right 
angle ; hence QF is perpendicular to CF, and Q lies always in a 
straight line through F, which is at right angles to CF. 

Hence the point Q moves in a straight line. 



Compound Cylinder Engine. 145 

It is universally admitted among scientific men that this is 
a discovery of the highest value as a contribution to the science 
of geometry, and the student will do well to examine it carefully 
in detail. For this purpose he should take the combination when 
unfettered by the bar DE, and should establish by trial the rela- 
tion between the sides, viz. : 

CDxCF = CR 2 -RD 2 . 

It is usual to call the figure FRDS a cell, the points D and F 
being the poles of the cell, and the arm ED being introduced 
simply to control the motion. 

Thus, if the pole D describes a circle of any given radius round 
some point in the direction of DC, the pole F must of necessity 
describe another circle whose centre lies also in the same line it 
may be that these fragments of circles have their convexities in the 
same or in opposite directions. All that is a matter for trial or 
study. And, again, the relative sizes of the circles will vary, so 
that it becomes possible to draw an arc of a circle of almost any 
required radius. 

The case in the text is where the two circular arcs, which 
always coexist when ED is centred at some point E in DC, or 
that line produced, are so related that the radius of one of them 
becomes infinite. 

ART. 122. In modern engines, the principle of the expan- 
sive working of steam is extensively carried out, and there are 
often two steam cylinders in the place of one, viz., a high-pressure 
cylinder of small dimensions, and a larger, or low-pressure cylin- 
der by the side of it. 

The steam is first admitted into the smaller cylinder, passes 
from thence into the larger one, and finally escapes into the con- 
denser, as in an ordinary condensing engine. 

The use of these double cylinders, with a swinging beam, 
necessitates a more complex form of parallel motion, but it is 
quite easy to understand the construction if we remember the 
principles already investigated. 

The diagram, fig. 164, shows the arrangement of the Pump- 
ing Engine at the Lambeth Water Works, where four engines of 
150 H.P. are placed side by side, arranged in two pairs, each pair 

L 



146 Elements of Mechanism. 

working into one shaft, with cranks at right angles and a fly-wheel 
between them. The stroke of the crank is equal to that of the 
larger cylinder, but the smaller cylinder, which receives steam 
direct from the boiler, has a shorter stroke, and its effective capa- 
city is about one-fourth that of the large or low-pressure cylinder. 
The peculiarity of the engines consists in the use of a crank and 
fly-wheel for controlling the motion of the pistons. The main 
pumps are connected with the beam near the end R. 

FIG. 164. 




The first part of the parallel motion for connecting the two 
piston rods with the beam of the engine consists of the portion 
CDBA, C being a fixed centre of motion, and A being the axis of 
the beam. In this portion the two arms, CD, AB, and the link 
BD give the parallel point P. 

If we now join AP, and produce it to meet the sides cf the 
superadded parallelograms in the points E and L, we shall obtain 
two other points whose motion is similar to that of P. To these 



Multiple Straight Line Motion. 



points the ends of the piston rods must be attached, and the 
arrangement is complete. The addition, therefore, of an inter- 
mediate bar, FEK, parallel to the side ML of the ordinary 
parallelogram, gives us what we require. < 

ART. 123. So likewise, in the parallel motion of Peaucellier, ' 
the superposition of the pantograph will give two or more points 
describing straight lines. 

Taking the ordinary combination, produce the lines CR, CS 
to the points H, L, making CH=CL, and add two bars, HT, LT, 

FIG. 165. 




of such length that they are respectively parallel to RP and SP. 
This parallelism must necessarily continue throughout the motion, 
and it follows that if PQ were drawn parallel to HR we should 
have an actual pantograph connecting P and T, just as Wat 
made it for a beam engine. Hence P and T must describe similar 
paths, which in this case are straight lines. . 

T 



148 



Elements of Mechanism. 



CHAPTER IV. 

ON THE CONVERSION OF RECIPROCATING INTO CIRCULAR- 
MOTION. 



ART. 124. It has been shown that the motion of a point 
in a circle results from the combination of two reciprocating 
movements technically called simple harmonic motions which 
take place in straight lines at right angles to each other, and that 
no single harmonic or other straight line motion can produce it. 
Also, that circular may be converted into reciprocating motion by 
the suppression of one of the above movements. 

The reconversion ot reciprocating into circular motion is not 
a problem of the same kind, as we now require the creation of a 





movement, instead of its suppression. Such a creation is impos- 
sible in a strict mathematical sense, but is practically attainable by 
mechanical construction. 

We recur to the contrivance of the crank, CP, and connecting 
rod, PQ, as one of the most obvious methods of solving the pro- 
blem ; it being clear that the travel of Q in a line pointing to C 
will cause the rotation of CP, and will compel P to move in a 
circular arc. 

But unless CP possesses inertia, or is attached to some heavy 
body as a fly-wheel, which, when once set in motion, cannot sud- 



Conversion of Motion. 149 

denly come to rest, there will be two points where the power 
exerted at Q will fail to continue the motion, and these points are 
evidently at A and B, where CPQ straightens into a right line. 

It is usual to call A and B the dead points in the motion, and 
P must be made to pass through them without deriving any aid 
from Q. 

In the existence of these dead points we note the failure of the 
contrivance as a piece of pure mechanism, and theory would tell 
us beforehand that it must fail, because we cannot create motion 
by mechanism any more than we can create force : we may 
modify or interchange without limit the movements which exist 
among the parts of a system, but there our power ends. And, 
accordingly, the continuous rotation of CP can only be provided 
for by storing up in the arm itself, or in some body, such as a fly- 
wheel, connected with it, the energy which is necessary to over- 
come any external resistance during that portion of the movement 
where the driver ceases to act. 

So, therefore, in applying the crank and connecting rod to 
beam-engines, the piston rod is attached by Watt's parallel motion 
to one end of a heavy iron beam, and the rotation of the fly-wheel 
is derived by the aid of a connecting rod or spear uniting the 
other end of the beam with a crank which turns the fly-wheel. 

The application to direct-acting engines has been already 
noticed, and the student will now understand that the mechanical 
working would be incomplete unless the crank were attached to a 
fly-wheel or other heavy revolving body balanced upon its centre, 
which would carry P through those portions of its path near to the 
dead points, and would also act as a reservoir, into which the 
work done by the steam might be poured, as it were, unequally, 
and from which it might be drawn off uniformly, so as to cause the 
engine to move smoothly and evenly. 

In marine engines, where the fly-wheel is not admissible, and 
where the engine must admit of being readily started in any posi- 
tion, two separate and independent pistons give motion to the 
crank shaft. In this case the two cranks are placed at right 
angles to each other, so that when one crank is in a bad position 
the other is in a good one. The same plan holds in the construc- 
tion of locomotive engines. 



1 50 Elements of Mechanism. 

ART. 125. In the instances considered, the circular motion 
derived from the reciprocating piece is continuous ; it now remains 
for us to examine a class of contrivances for producing the like 
result where the circular motion is intermittent. 

The circular motion being that of a wheel turning upon its 
axis, it may be arranged that one-half of the reciprocating move- 
ment shall be suppressed, and that the other half shall always push 
the wheel in the same direction. This is the principle of the 
ratchet wheel. 

Or it may be arranged that the reciprocating piece shall be of 
the form shown in those escapements which produce a recoil, and 
the pallets will then act upon opposite sides of the wheel, and will 
drive it always in one uniform direction. Here we find ourselves 
again in the region of pure mechanism. These two classes of con- 
trivances constitute the only methods by which the problem can 
be solved without external assistance. 

ART. 126. A wheel provided with pins or teeth of a suitable 
form, and which receives an intermittent circular motion fr< 
some vibrating piece, is called 
ratchet wheel. 

In the drawing E represents th'e 
ratchet wheel furnished with teeth 
shaped like those of a saw, and AB, 
the driver, is a click or paul, jointed 
at one end, A, to a movable arm AC, 
which has a vibrating motion upon 
C as a centre. 

As AC moves to the right hand, 
the click, B, pushes the wheel be- 
fore it through a certain space. Upon 
the return of AC, the click, B, slides over the points of the teeth, 
and is ready again to push the wheel through the same space as 
before, being in all cases pressed against the teeth by its weight 
or by a spring. 

A detent, D, prevents the wheel from receding while B is 
moving over the teeth, for it is, of course, a condition in this move- 
ment that the ratchet wheel itself shall either tend always to fly 
back, or shall remain held in its place by the friction of the pieces 





Ratchet Wheel. 



FIG. 168. 



with which it is connected. In this way the reciprocating move- 
ment of AB is rendered inoperative in one direction, and the 
circular motion results from the suppression of one half of the 
reciprocating movement of the arm. 

The wheel, E, and the vibrating arm, AC, are often centred 
upon the same axis. 

ART. 127. As regards the action between the teeth and the 
detent, we observe that the wheel must tend to hold the detent 
down by the pressure which it exerts, 
and that it will do so as long as the 
line of pressure on the surface pr 
falls below the centre D. 

If the angle qrp were opened 
out much more, the perpendicular 
upon pr might rise above D, and 
the detent would then fail to hold 
the wheel. 

Further, the click has to return 
by slipping over the points of the teeth. The condition for this re- 
sult is that the perpendicular to the surface qr shall fall between 
D and the centre of the wheel. 

Where very little force is required to hold the wheel, and the 
exact position is of consequence, as in counting machinery, the 
teeth may be pins, and the detent may be a roller pressed against 
them by a spring. 

The usual form of the teeth is that given in fig. 167, and the 
result is that the wheel can only be driven in one direction ; but 

FIG. 170 






in machinery for cutting metals it is frequently desirable to drive 
the wheel indifferently in either direction, in which case a different 
construction is adopted. The ratchet wheel has radial teeth, and 



152 



Elements of Mechanism. 



the click, B, can take the two positions shown in the diagrams, 
and can drive the wheel in opposite directions. 

Here the click has a triangular piece fitted upon its axis, any 
side of which can be held quite firmly by a flat stop attached to a 
spring. There are, therefore, three positions of rest for the click, 
whereof two are shown in the figure, and the third would be found 
when the click was thrown up -in the direction of the arm pro- 
duced. We have by this simple contrivance a ready means, not 
only of driving the wheel in both directions, but also of throwing 
the click out of gear when required. 

ART. 128. A common method of applying a ratchet wheel in 
ordinary mechanism is to be found in the drawing of a screw-jack, 
as given at the end of the first chapter. (See fig. 38.) 

The object of the ratchet there sketched is to drive the screw 
which traverses the framework of the jack, and it will be readily 
understood that the friction on the 
screw thread is so great that there is no 
tendency for the wheel to run back, and 
that no detent is necessary. 

In fig. 171, the lever handle, EH, 
which actuates the driving paul, has a 
centre of motion coincident with that 
of the ratchet wheel, and the paul D is 
held against the teeth by a spring S. 
On moving the handle to the right the 
paul slips over the teeth, and on moving 
it to the left the spring presses the paul 
upon the teeth, and the ratchet is ad- 
vanced in the usual way. 

The sketches show the lever-handle 
and ratchet both in front and side ele- 
vation. 

This is the same contrivance as an ordinary ratchet-brace used 
in drilling by hand. 

ART. 129. Everyone must have seen the application of the 
ratchet wheel to capstans and windlasses, where it is introduced 
in order to prevent the recoil of the barrel, the same purpose for 
which it is applied in clocks and watches. 





Multiple Pauls. 1 5 3 

It was a very early improvement to provide two pauls of differ- 
ent lengths, termed by the sailors paul and half paul, and thereby 
to hold up the barrel at shorter intervals during the winding on of 
the rope ; in fact, a ratchet wheel of eight teeth thus became 
practically equivalent to one of sixteen teeth, and the men were 
better protected from any injury which might be caused by the re- 
coil of their handspikes. 

The principle of this contrivance is very intelligible, and is 
shown in fig. 172, where the two pauls DP, EQ, differ in length by 
half the space of a tooth. 

As the wheel advances by in- 
tervals of half a tooth, each paul 
falls alternately, and the same effect 
is produced as if the number of 
teeth were doubled, and there was 
one paul. 

In the same way three pauls 
might be used, each differing in length by one-third of the space 
of a tooth, and so the subdivision might be extended. 

In practice we rnay be required to move a ratchet wheel 
through certain exact spaces, differing by small intervals. Where 
such is the case it is better not to attempt a minute subdivision of 
the teeth, as they become liable to break and wear away and the 
action is uncertain, but recourse should be had to a method of 
placing three or four clicks upon the driving arm. 

In such a case the pauls or clicks will be increased in number, 
and will act as drivers instead of 
detents, being arranged in order of 
magnitude as regards length. They 
may be placed upon separate driving 
arms, but there is no advantage in 
doing so, and it is usual to place 
them all upon one pin at the end of 
the driving bar. 

In fig. 173, two clicks, DP, DQ, are shown hung upon a pin D, 
which is supposed to be at the end of the arm which drives the 
wheel The clicks differ in length by half the space of a tooth, and 
they will manifestly engage the wheel alternately, and will move it 




154 Elements of Mechanism. 

as if there were twice as many teeth driven by one click. And so 
for three or any greater number of clicks. 

On referring back to fig. 53, the student will find an example 
of the use of a ratchet wheel. A link, HK, connects the recipro- 
cating frame, FG, with an arm, LM, carrying a click at Q ; thus 
the oscillations of the frame are received by the arm, and the 
wheel is advanced a certain number of teeth upon each motion to 
the right. The number of teeth taken up can be regulated by 
adjusting the distance of K from L by means of a screw ; the 
nearer K is brought to L, the greater will be the advance of the 
ratchet wheel at each stroke. 

The object of the arrangement is to feed on the wire of lead 
from which the material for each bullet is cut, and by placing 
three clicks at Q instead of one, according to the method just 
examined, the amount of advance for bullets of different sizes may 
be regulated with considerable nicety. 

As, in some form or other, the principle of this construction is 
of great practical value, we will examine it more in detail. It will 
be found that one chief use of the ratchet wheel occurs in provid- 
ing the feed motions in machinery for cutting or shaping metals, 
and the general plan adopted is to draw off, as it were, some 
definite amount of motion at proper intervals during the opera- 
tion, and so to impart an unchangeable movement of vibration 
through a definite angle to a bar which comes as the first piece on 
the way to the ratchet wheel. We stari by giving to such a bar 
some fixed artd unvarying amount of vibration, and our object 
will be to draw off from this motion just so much or so little as we 
may require for the purposes of the work, and so to advance the 
ratchet by i, 2, 3, or any convenient number of teeth. 

If we can now arrange to do this, we have obtained the first 
part of a feed motion, viz., the power of advancing the ratchet at 
each stroke by an integral number of teeth ; but we may want to 
go further, and advance the wheel with greater nicety, such as 
through 2\ of a tooth each time, and the student will under- 
stand that for the integral part of the advance some definite con- 
trivance, such as those we are about to discuss, will be wanted, 
and that for the fractional part of the advance this system of 
multiple clicks will be perfectly sufficient, 




Feed Motion. 1 5 5 

ART. 130. In considering this problem of advancing the 
ratchet through any required number of teeth, the very obvious 
principle already suggested will become apparent upon inspecting 
the diagram. 

Suppose the arm AC to represent a vibrating bar which swings 
with a definite amount of angular motion through the space PCQ. 
In a planing machine 
this bar would be pushed 
over each time that the 
table came to the end 
of its travel. If AC 
be connected to another 
point S by the link RS, 
we can impart to the 
point S a reciprocating 

motion in the line CS, which may be represented by the space FE 
under the conditions shown in the sketch. It is now clear that if 
R be moved towards C, this travel, FE, will diminish to nothing, 
while AC continues all the time to swing through the same angle 
PCQ, whereas FE will be increased in a like degree when R is 
moved away from the centre C in the direction Rr. 

Thus the rate of advance of the ratchet wheel may be regu- 
lated. 

ART. 131. We have seen that when a bar centred at one 
end is continually moved to and fro through a given angle, a teed 
motion can be derived from it, and it will be found in the work- 
shop that all kinds of rough but effective contrivances are adopted 
for obtaining such a motion. 

We subjoin an elaborate arrangement taken from a rifling 
machine in the Arsenal at Woolwich. 

The student should refer back to Art. 67, where he will find 
an account of the head of a rifling bar, and will note an arrange- 
ment for pushing cut the cutters when the head of the rifling bar 
reaches the breech of the gun, and for withdrawing the cutters 
while the head is being advanced into the bore. In other words, 
he will note that the cut is made while the head is being drawn 
out. 

It will also be seen in fig. 80, that when a piece marked RS 



I 5 6 



Elements of Mechanism. 



is pulled back, the cutters enter the work, and when RS is pushed 
forward, the cutters are sheathed within the head. It follows that 
some contrivance must be adopted for moving RS longitudinally 
through a small space at each end of a stroke. 

In our present lecture diagram the piece to which we have re- 
ferred is not shown, for it lies at some distance to the left beyond 
the range of the drawing, and forms the head of the rifling bar 
TT. In the previous diagram the end of the bar TT, which is a 
hollow pipe, is shown in section, and the student must conceive 
that a slender rod passes from the head of the rifling bar along the 
whole length of TT, and terminates in the piece marked A in the 
annexed sketch. 

The whole operation, with which we are now concerned, con- 
sists in moving A longitudinally to and fro through a small space. 
If that can be done at the right times, the feed motion is com- 
plete. 




In the drawing there is a lever CR, having a fulcrum at C, and 
this lever terminates in a short arm concealed within the saddle 
S, which moves A when CR is turned through a definite angle. 
The movement of CR is determined by causing the end R to rest 
either upon bb or aa. When R is on bb the piece A is pulled out, 
and the cutters are at work; whereas, when R is on aa the piece A 
is pushed in, and the cutters are sheathed. 

Conceive that the cut is being made, that R is travelling to 
the right along bb, and that the roller which is loaded by a weight 
P reaches FL. The piece FL is a bar jointed at F, and rises out 
of the way to allow R to pass. As soon as it has done so, and FL 



Feed Motion. 157 

has dropped into its place, the cut is complete, the mechanism of 
the machine is reversed, the rifling head begins to enter the gun, 
the saddle carrying CR moves to the left, the roller R travels up 
the incline, and the lever CR is raised so that R rests upon aa, at 
which time the piece A is pushed forwards, and the cutters are 
sheathed. 

The roller R travels along aa until it reaches the weighted 
drawbridge D, which it overpowers, and so travels on to the part 
marked E. When it gets there, the rifling head has reached the 
breech end of the gun, and the mechanism is reversed. As the 
saddle returns the drawbridge has risen, and R runs down the 
slope on to its former path bb. Thus R travels to and fro along 
bb and aa alternately, and the necessary feed motion is obtained. 

ART. 132. A feed motion, extremely ingenious in principle, 
has been applied in Sir J. Whitworth's planing machines, and is 
worth a careful examination. In fig. 176, let CA represent a vi- 
brating bar centred at C, upon which point there is also centred a 
circle carrying two pins, P and Q. 

We will suppose that the circle vibrates, 
independently of the arm, through an angle 
exactly represented by PCQ, and that the 
object of the contrivance is to impart to 
the bar, CA, the whole or any portion of 
this vibration. 

It is easy enough to impart the whole 
vibration. We have only to fasten P and 
Q close to CA on each side of it, and the 
bar and the circle will swing as one piece. 

Again, if we wish the bar to remain at rest, we may separate P 
and Q as much as possible, and when the bar has been pushed 
as far as it can go by one of the pins, it will remain at rest, for the 
second pin can only just come up to it on the opposite side. 

Conceive now that the pin Q is shifted to q, the arm CA will 
be pushed to Q by the pin P when moving to the right, but will 
only return as far as Q can carry it, i.e., to q, and the vibration 
will take place through an angle QQr instead of an angle QCP, 
and in this way, by separating P and Q, the motion of AC may be 
reduced till it ceases altogether. 




1 5 8 



Elements of Mechanism. 



FIG. 177. 




So, therefore, we obtain precisely the same result as in a 
previous case, and can advance the ratchet wheel through any in- 
tegral number of teeth up to a limit 
fixed by the amount of the vibration of 
the arm. . 

The practical arrangement is shown 
in fig. 177, where a ratchet wheel, an 
arm carrying a click, and another wheel 
provided with a circular slot, are placed 
in the order stated upon the same axis, 
and can all move independently of each 
other. There are two movable pins in 
the circular slot, which are capable of 
being fixed in any position by nuts at the back of the wheel, and 
which embrace the arm carrying the click, but do not reach the 
click itself. 

The ratchet wheel is connected with a screw which advances 
the cutter across the table, and the object is to impart definite but 
varying amounts of rotation to the screw after each cut has been 
taken. 

The wheel F receives a fixed amount of vibration from the 
table, and will impart the whole thereof to the click if P and 
Q be made to embrace the arm closely upon each side ; or it will 
impart -any less amount, gradually diminishing to zero, as P and Q 
are separated to greater intervals along the groove, and thus the 
feed of the cutter may be regulated. 

ART. 133. A mechanical equivalent for the teeth and click 
may be found in what is termed a nipping lever, constructed upon 
the following principle. Conceive that 
a loose ring B surrounds a disc A, and 
that upon a projecting part of the ring 
there is a short lever, DE, centred. 
This lever is movable about a fulcrum 
at F near to the wheel, and terminates 
at one end in a concave cheek, D, fitting 
the rim of the disc. On applying a 
force at E the lever will nip or bite upon the disc, and the friction 
set up may be enough to cause them to move together as if they 



FIG. i 




, 



Silent Feed. 159 

were one piece. This friction does, in fact, increase almost in- 
definitely, for the harder you pull at E the greater will be the 
pressure at D, and since the friction increases with the pressure, 
being always proportional to it in a fixed ratio, the resistance of 
friction will be developed just as it is wanted. 

Upon reversing the pressure at E, the nipping lever will be 
released and the ring will slide a short space upon the disc : thus 
the action of a ratchet wheel is imitated. 

ART. 134. The ratchet wheel has been much used in ob- 
taining an advance of the piece of timber at each stroke of the 
saw in sawing machines. A substitute has been found in an 
adaptation of this nipping lever, and is commonly known as the 
silent feed. 

An arm AB, centred at C, rides upon a saddle which rests 
upon the outer rim of a wheel ; a piece, EE, is attached to one end 
of the arm, and admits of being 
pressed firmly against the inside of 
the rim of the wheel which carries 
the saddle. 

It is clear that when the end, 
B, of the arm, ACB, is pulled to the 
right hand, the rim of the wheel will 
be grasped or nipped firmly between 
the saddle and the piece EE, and 
that the pull in BD will move the 
saddle and wheel together, as if they 
were made in one piece. When B 
is pushed back, a stop prevents BCA 
from turning more than is sufficient to loosen the hold of EE, and 
the saddle slides upon the rim through a small space. - 

In this way the action of a ratchet wheel is arrived at, and, by 
properly regulating the amount of motion communicated by the 
link BD, we obtain an equivalent for a ratchet wheel, with an in- 
definite number of teeth. 

It is this circumstance which renders the contrivance so useful. 
The amount of feed of the timber can be regulated with the utmost 
nicety ; the opposite end of the link BD is movable by a screw, 
according to the principle laid down in Art. 130, and can be set at 




i6o 



Elements of Mechanism. 



any distance from the centre of the arm upon which it rides, and 
thus the contrivance is as perfect as can well be imagined. 

A screw, F, may be employed to bring up a stop, H, towards 
the arm, ACB, and so to prevent the arm from twisting into the 
position which gives rise to the grip of EE. The saddle will then 
slide in both directions without imparting any motion to the wheel, 
a result which is obtained in an ordinary ratchet wheel by throwing 
the click off the teeth. 

ART. 135. A like result may be obtained in a more simple 
manner. The annexed drawing of Worssam's feed motion has 
been taken from a model belonging to the School of Mines. 

FIG. iSo. 




The wheel B is the feeding wheel, and the object is to impress 
upon it a step-by-step motion changing in every possible amount 
between two fixed limits. 

A handle turns the crank CP, whereby P performs complete 
revolutions, and the student will recognise the well-known combi- 
nation of CP with a second crank BQ and the connecting rod PQ, 
the result, of course, being that while P describes a circle round 
C, the arm BQ oscillates through a given angle. There is a con- 



Silent Feed. 161 

trivance for adjusting the end P of the connecting rod PQ at 
different distances from C, and indeed for bringing P to coincide 
with C. The consequence of this arrangement is that BQ will 
continually swing through a less and less angle, diminishing to 
zero, as P is brought nearer coincidence with C. 

At the end D of the arm QB produced there is a circular piece 
which is shown on an enlarged scale in the detached sketch, and 
is there marked F. The object of drawing this piece a second 
time is to make it quite clear that the circular rim F is eccentric 
to the spindle on which it rides. In the drawing the centre of the 
spindle is marked O, and the centre of the circular rim F is 
marked n. 

A second detached sketch shows the rim of the wheel and the 
piece F in section, from which it appears that F is a circular wedge 
fitting into a corresponding angular recess which runs round the 
whole rim of the wheel. The object of making F eccentric is now 
manifest, for as F rotates on its spindle towards the left hand, the 
wedge is driven more tightly into the recess and the surfaces grip 
together with great holding power. As soon, therefore, as BD turns 
through a definite angle in the direction of the arrow, the piece F 
holds to the wheel just as if it were a driving paul engaging with a 
ratchet tooth, and of necessity the wheel moves forward. On re- 
versing the motion of BD the grip is released, and F slides back 
over the rim without any opposition. In like manner the upper 
piece corresponding to F acts as a detent, and allows the wheel to 
advance in the direction of the arrow but stops any return. By 
this simple construction we have a driving paul and a detent fitted 
to a wheel having any required step-by-step movement as deter- 
mined by the angular motion imparted to BD. 

ART. 136. Where the ratchet wheel moves at each vibration 
of the driver, and not during every alternate movement, an escape- 
ment, or something approximating thereto, must be ' employed. 
The action now takes place alternately upon opposite sides of the 
wheel. 

Upon looking back to the elementary form of escapement 

described in Art. 49, it is quite apparent that if the frame AB be 

moved to the left, the pallet D will push P before it, sufficiently 

far to bring R in front of C, and then, upon the return of the 

M 



1 62 Elements of Mechanism. 

frame, the pallet C will push R before it, and thus the scape wheel 
will rotate always in one direction. 

We remark that this direction is the opposite to that in which 
the wheel revolves when driving the escapement. 

The same thing is true, generally, of all Recoil Escapements, 
and upon examining them it will be found that the scape wheel 
may be driven backwards by the pallets. 

ART. 137. A like action results where two clicks are hung 
upon a vibrating bar, and one of them terminates in a hook. 

The bar ECD vibrates on C as a centre, and the pieces QD, 
PD, hang at the extremity D. (Fig. 181.) 

When P pushes on the wheel, the arms PD and DQ open, and 
the hook at Q slips over a tooth : whereas, upon reversing the 
motion of the driving lever, the hook drags the wheel with it, and 
P slips over a tooth, and thus the wheel advances upon each 
vibration of the moving arm. 

This contrivance is due to Lagarousse. 





The click may be replaced by a hook turned in the reverse 
direction, as in the annexed example, which is taken from spin- 
ning machinery (fig. 182). 

Here the bell crank lever ECD is furnished with the hooks 
BP, DQ, and by swinging it to and fro through the angle EO, we 
shall catch up a tooth at the points P and Q alternately, and shall 
drive the wheel in one direction. 



Levers of Lagarousse. 



1 63 



The hooks may both be replaced by pauls turned in the reverse 
direction, as in the annexed example, which is taken from a model 
belonging to the School of Mines. 




Here the lever handle CE swings to and fro through an arc 
Ee, and operates upon the pauls P and Q in the manner indicated. 
The motion of CE is restrained by a stop S, and when CE is 
being raised as far as it will go, the paul Q is gathered up over 
the teeth, while the paul P urges the wheel B onward. In like 
manner, when CE is depressed, the paul Q drives, and P is lifted 
upwards over the teeth. Thus the wheel advances both when CE 
rises and when it descends, and it will be found that it moves 
through two teeth at each stroke. The piece D is an ordinary 
detent for preventing the recoil of the wheel. 

ART. 138. The construction of mechanism for numbering, or 
printing consecutive numbers, has wonderfully advanced in late 
years, and many ingenious improvements have been originated. 
We do not intend at present to enter upon any details, but there is 
one contrivance known as a masked wheel which ought to be 
understood. 

The object of the masked wheel is to enable a numbering 
machine to print the same number twice before the unit advances, 
as in numbering a cheque and its counterfoil. 

The masked ratchet wheel consists of two ratchet wheels placed 
side by side, the teeth in one being the ordinary uniform teeth, 
while the teeth in the other are alternately shallow and deep. The 



1 64 Elements of Mechanism. 

main feature is that the bottom of the cut in a shallow tooth is 
sufficiently removed from the centre to cause the driver of the 
ordinary wheel to be raised entirely above the teeth of the same, 
whereby it follows that when the paul rests in the shallow cut it 
does not operate on the teeth in the other wheel. When the paul 
rests in the deep cut it acts as a driver to both wheels. 

The two wheels ride upon the same axis, but move indepen- 
dently of each other, the ordinary ratchet wheel being connected 
with the numbering apparatus. 

In the drawing, which is taken from an old model belonging to 
the School of Mines, a pin wheel represents the first ratchet wheel, 
and the second wheel C has its teeth in pairs, every alternate tooth 
being cut deeper in the manner stated. 

FIG. 184. FIG. 185. 





The paul PD has a driving pin Q, which must always engage 
with a tooth of the wheel C. but which may or may not drop low 
enough to take up a pin on the other wheel. 

In fig. 184 the paul has engaged a shallow tooth in the wheel 
C, and is ready to drive that wheel onwards ; but the paul has its 
point p clear of the pins on the shaded wheel, being masked or 
prevented from acting by the shallowness of the cut at Q. 

After the paul has driven on the wheel C by the space of a 
tooth, it will next fall into the deeper cut, when both Q and / will 
engage their respective teeth, and the two wheels will advance 
together in the manner shown in fig. 185. 

Thus two strokes of the paul are required in order to advance 
the pin wheel by one tooth. 

We subjoin two sketches of a masked ratchet wheel as con- 
structed in a small numbering machine. 



Masked Ratchet. 



[65 



The wheels are shown in elevation, one behind the other, and 
having a common axis at C. The unshaded disc represents an 
ordinary ratchet with uniform saw-shaped teeth, whereas the tinted 
wheel has deep and shallow teeth alternately. In the machine 
referred to the method of advancing the ratchet is peculiar, the 
driving paul is stationary, and the axis of the ratchet wheels is 
swung round parallel to itself in a circular arc, as indicated by the 
dotted line af. 




The driving paul is in the form of a lever, DAP, having a ful- 
crum at A and a projection at P, which engages with the teeth of 
the ratchets. The shape of the end of the lever is something 
like a boot, and the pointed end acts as a driver under other con- 
ditions, with which we are not concerned at present. An elastic 
india-rubber band, DE, presses the paul P into contact with the 
ratchets. 

As the wheels sweep to and fro from/ to #, and back again, 



i66 



Elements of Mechanism. 



FIG. 187. 



the piece P catches the masking or shaded wheel, and moves it 
on through the space of one tooth. On the next stroke the piece 
P falls upon a deeply cut space in the masking wheel, and being 
broad enough to engage both wheels, it carries them on together 
through the space of a tooth. Thus the unshaded ratchet is 
moved through the space of a tooth at each alternate stroke in the 
direction of the arrow. 

ART. 139. The converse of the contrivance described in 
Art. 68 may also be used to convert reciprocating into circular 
motion; that is to say, the bar may be employed to turn the screw 
barrel, instead of the screw driving the bar : but such an arrange- 
ment gives rise to a great increase of friction, and is only met with 
when a small amount of force is to be exerted. 

There is a well-known instance in light hand drills, called 
Archimedean drills, where the rotation is 
derived by pushing a nut up and down a rod, 
upon which a screw of rapid pitch is formed, 
the drill rotating in opposite directions as the 
nut rises and falls. 

This movement was at one time proposed 
by Sir J. Whitworth, in order to obtain a 
reversing motion of the cutter in planing 
machinery. 

Here a rod, HG, was provided with a sort 
of tooth, G, which fitted exactly into a groove 
in the form of a screw-thread traced upon the 
cylinder AB. As the rod moved up and 
down it reversed the position of the cutter, 
and enabled it to act while the table was moving in either 
direction. 

This reversing motion has not been used, but another has been 
employed in the place of it, where an endless catgut band, which 
runs round a pulley capable of vibrating through a given angle, is 
carried on to a small pulley attached to the top of the tool box, 
the cutter turning through two right angles when the band is made 
to traverse a small space in opposite directions by the action of 
the vibrating pulley. 

This band is kept stretched by a tightening pulley, and a 




Ratchet and Pump. 167 

second turn round one or more pulleys in the circuit will always 
prevent the band from slipping. 

ART. 140. When the diameter of a ratchet wheel is increased, 
the curvature of the rim is diminished, and the ultimate form of a 
portion of the enlarged wheel is that of a straight bar. Such a 
ratchet bar is often useful, and we have seen an example in the 
machine for cutting conical boxwood plugs. 

It has long been observed that every movement which is con- 
strained to take place in one direction, and cannot be reversed, is 
that of a ratchet wheel or bar. Accordingly, the analogy between 
a common suction pump and a ratchet bar is complete. When 
the pump is at work the water must rise, and can only descend by 
some leakage or imperfection. 

The mechanism of a pump consists of a bucket with a valve 
opening upwards, which we may call A, and a fixed valve B, in 
the pump barrel, which also opens upwards. Our object is to 
point out to the student that the column of water passing through 
the pump may be viewed as a vertical ratchet bar, that the driving 
paul is the same thing as the pump bucket and valve A, while the 
detent is the lower valve called B. 

It is clear that the valve B acts as a detent, for it allows the 
water to pass freely upwards, but will not permit its descent. It 
is equally clear that the bucket and valve A act as a driving paul, 
for during the descent of the bucket the valve A opens, and the 
hold upon the column of water is obtained at a lower point. The 
action is precisely that of a driving paul when slipping over the 
teeth of a bar, and holding on at a lower point. On the ascent 
of the bucket the column is lifted just as the driving paul lifts a 
rigid ratchet bar. 



1 68 Elements of Mechanism. 



CHAPTER V. 

ON THE TEETH OF WHEELS. 
& 

ART. 141. We propose now to enter into a mathematical 
/^ investigation of the forms of teeth adapted for the transmission 
of motion or force in combinations of wheelwork, and we have 
already stated the general nature of the problem. 

It is required to shape the teeth or projections upon the edges 
of two circular discs in such a manner that the motion resulting 
from the mutual action of the teeth upon the discs or wheels shall 
be precisely the same as the rolling action of those definite circles 
known as the pitch circles of the wheels in question. 

Commencing with flat plates or discs, which, in the case of 
light wheels, such as are used in clockwork, would be castings in 
brass or gun metal, having light rims of sufficient substance to 
allow of the cutting away of the material so as to shape the teeth, 
we should settle in the first instance the exact size of the pitch 
circles, and next the number and pitch of the teeth which we 
meant to employ. 

The last inquiry involves only a very elementary knowledge of 
geometry : we have merely to find out how many times we can 
repeat the space occupied by the pitch of a tooth upon a circle of 
given diameter, so as quite to fill up the circumference without 
any error in excess or defect. 

ART. 142. It will be seen at once that the following very 
simple equation connects the diameter of a pitch circle with the 
number of teeth and their pitch. 

Let D be the diameter of the pitch circle in inches, 
P the pitch of a tooth in inches, 
n the number of teeth upon the wheel. 

Then nP = circumference of pitch circle = rD, 
where * = 3-14159, or = 2 T 2 approximately. 

Hence n = x D, or D = x n. 



Teeth of Wheels. 



169 



In order to save trouble, definite values are assigned to P, 
such as \, f, \, |, f, |, i, 1 1, i|, i, 2, 2\, &c., and the values of 



~, , are calculated and registered in a table of which we give 
r TT 


specimen. 






^ 


p 






p 


JT 




2 


i- 57 c8 


6366 




2* 


1-3963 


7l62 




i 


i -2566 


7958 




2| 


i-'333 


8754 




3 


i -0472 


'9549 



Thus, let P = 2 Jj inches, we find in the table that 
~= i -2566 and = 7958. 

Suppose a wheel of 88 teeth, and 2^ inch pitch, to be in course 
of construction, and that we require to know the diameter of the 
pitch circle. 



D =- x n ---- 7^58 



70-03 inches. 



Or, again, if the diameter of the pitch circle be 70 inches, and 
the number of teeth of 2^-inch pitch be required, 

n = ^x D= 1-2566 x 70 = 87-96=88 very nearly. 

In practice it is more easy to treat of the subdivision of a 
straight line than of the circumference of a circle, and it is the 
custom therefore to suppose the diameter of the pitch circle to 
be divided into as many equal parts as there are teeth upon the 

wheel, and to designate as the diametral pitch in cosHlpfc^dis- 
tinction to P, the circular pitch. / /y"~7 

Further, let ~= where m is a whole number, T~~ 

n m 

now D = P j: = P p=J ,. 
n TT m TT m 



I/O 



Elements of Mechanism. 



The values of m and are registered in a table, of which a 
m 

portion is given, and the circular pitch can be at once found. 



Values of m 


3 


4 


S 


6 


7 


8 


9 


ro 





Values of P 


1-047 


785 


628 


524 


'449 


'393 


'3*9 


3M 


262 


or approximately 


I 


I 


1 


i 

2 


1G 


f 


ft 


j'c 


1 

4 



Thus, let D = 8 inches, n = 80, 



_ 

80 10 

Hence P '314= {' G inch. 

ART. 143. A very simple piece of apparatus, called a sector, 
is constructed to save workmen trouble in arranging the sizes of 
wheels with given numbers of teeth to work with given pinions or 
conversely. 

It consists of two light straight arms of brass, centred upon a 
pivot, with an arc of a circle at the end of one of the arms, and 
a binding screw to clamp the other arm to the arc. Thus the arms 
can be set at any angle. These bars are each graduated at intervals 
along their length, the intervals being equal except for some of the 
smaller numbers up to 10, and the reading may extend to 150 or 
thereabouts. 

Conceive now that a pinion with 12 leaves or teeth is placed 
between the bars, and that they are closed up till the pinion just 
reaches the graduation 12, and let it be required to find the size 
of a wheel having 96 teeth to work with the pinion. We should 
measure the distance across the bars at the graduations 96, and 
that would give the required diameter. 

This is evident, because if a disc were placed between the bars 
with its plane perpendicular to the plane of the bars, its diameter 
would form the base of an isosceles triangle, of which the two 
legs form the sides, and the bases of any of these triangles must 
be in the same proportion as the respective sides, so long as the 
vertical angle remains unchanged. 



Teeth of Wheels. i? I 

Thus the readings of the legs give the proportionate diameters 
of the wheels and pinions, and hence also the proportionate num- 
bers of teeth in the same. 

ART. 144. As far as we have gone at present, we have simply 
determined the relation between the pitch of a tooth and the 
circle upon which it is formed : it will be necessary to consider 
also how much of this arc called the pitch is to be occupied by 
the solid tooth, and how much by the adjoining empty space : 
we must arrange also the depth of the tooth and the depth of the 
open space. But without touching upon these points at present, 
it may be convenient to work out the problem of shaping the 
teeth in such a manner that the wheels shall roll with perfect 
accuracy upon each other, precisely as the ideal pitch circles, 
which we have already referred to, would move by rolling contact. 

Two principal propositions on which we rely have already been 
demonstrated, viz. : 

1. When two circles roll together, their angular velocities are 
inversely as the radii of the circles. 

2. When two arms or cranks are connected by a straight link, 
the angular velocities of the arms will be inversely as the segments 
into which the direction of t/ie link divides the line of centres. 

ART. 145. The latter proposition' applies equally when FQ 
meets CB between the centres C and B, as in the case now to be 
dealt with ; and we must remind the student that geometers have 
shown that every curved line may generally at each point be sup- 
posed to possess the curvature which would be found in a circle of 
definite size called the circle of curvature of the point in question. 
In the modern editions of Newton's 'Principia,' the circle of cur- 
vature at any point of a curve is defined as being that circle which 
has the same tangent and ctirvature as the curve has at the point in 
question. 

It is also capable of proof that no other circle can be drawn 
whose circumference lies between the curve and its circle of curva- 
ture, starting from the point considered. 

In order to find this circle of curvature at any point P of a 
curve, we first draw the tangent at P, we then take a very small 
arc PQ of the curve tenninating in P, and from the other ex- 
tremity Q of the arc we draw QR perpendicular to the tangent at 



1/2 



Elements of Mechanism. 



P, and meeting it in the point R ; the diameter of the circle of 



curvature will then be the limiting value of the ratio 



(arc PQ) 2 




QR 

ART. 146. We must now assume that this circle can be 
determined, and in fig. 188 we will make A and B the centres of 
FIG. 188. motion of two pieces pro- 

vided with teeth of some de- 
terminate form, which are in 
contact at the point /. 

Draw now P/Q, a com- 
mon perpendicular to the 
surfaces of the teeth at / ; 
and let P and Q be the cen- 
tres of the circles of curvature 
of the curves which touch at 
p : draw also AR, BS, per- 
pendicular to PQ. 

Then, in the first instant 
of motion, PQ may be re- 
garded as constant, because it is the distance between the centres 
of two ascertained circles which do not vary in size for a very 
small amount of sliding motion of the curves upon each other ; 
and therefore the angular velocities of the two pieces will be iden- 
tical with those of AP and BQ. 
But we have just proved that 

angular vel. of AP_BS _BE . 
angular vel. of BQ~~AR~~AE ; 

, angular vel. of piece A BE 

whence -;*-. ^ = TT^- 

angular vel. of piece B AE 

In order to connect our investigation with this case of link- 
work motion, we have only to remember that an imaginary com- 
bination of the two arms BQ, AP, connected by a link PQ is 
always supplied, and that although the separate parts of this com- 
bination are continually changing, yet it is always present as a 
whole, and gives us the means of comparing the angular velocities 
of the pieces A/ and B/ at every instant. 

Suppose it to be required that the angular velocities of the 



2 eeth of Wheels. 173 

two pieces shall be the same as those of the pitch circles of radii 
AD, BD, which is the case in wheelwork. 

We must now form the curves so that E shall coincide with D, 
and shall never leave it during the motion ; in other words, the 
common perpendicular to the surfaces of any two teeth in contact 
must always pass through the point of contact of the two pitch circles. 

If the teeth can be formed so as to satisfy this condition the 
problem will be fully solved, and we proceed to give the solutions 
which have been devised by the ingenuity of mathematicians. 

There are two curves which will be of great assistance to us, 
which are the following : 

i. An epicycloid is a curve traced out by a point, P, in the 
circumference of one circle, which rolls upon the convex arc of 
another circle. This curve is represented by the dotted line in 
figure 189. 

FIG. 189. FIG. 190. 





2. A hypocydoid is a curve traced out by a point, P, in the 
circumference of one circle, which rolls upon the concave arc of 
another circle. This curve is represented by the dotted line in 
figure 190. 

ART. 147. Conceive now that we are about to form these two 
curves, the one an epicycloid upon the outside of the pitch circle 
A, the other a hypocydoid upon the inside of the pitch circle B : 
we will employ the same generating circle in both cases, and will 
suppose that A and B represent the pitch circles of two discs 
upon which teeth are to be carved out. Our object is to show 
that teeth shaped according to these curves will answer the purpose. 



174 



Elements of Mechanism. 




Having drawn the curves, bring the circles together till the two 
circles touch in the line of centres and the two curves in another 
FIG. 191. point P, as shown, when it will be 

found that the common perpendicu- 
lar at the point of contact of these 
curves passes through D. 

The truth of this statement will 
be evident from the consideration 
that when the curves touch, the 
describing circle may be taken as 
being ready to generate either the 
one curve or the other. Now the 
describing circle cannot do this un- 
less it be resting upon both circum- 
ferences indifferently, that is, resting 
upon the point where the circum- 
ferences of the pitch circles touch each other, in which case the 
common perpendicular to the curves at P passes through D. 

But this is the very condition which we are seeking to fulfil, 
because if it maintains the teeth will move the discs upon which 
they are shaped with a relative velocity, which is represented by 

BD 

the ratio --, and we have shown that the relative velocity of the 
Al_) 

two pitch circles is also - , hence the relative velocity of the 
AL) 

discs furnished with teeth is precisely the same as that of the 
pitch circles, and the problem which we are investigating is com- 
pletely solved. 

ART. 148. We have now obtained two curves which satisfy 
the geometrical requirements of the problem, and it remains to 
put our theory into practice. 

The epicycloid and hypocycloid which form the acting surfaces 
of two teeth must be produced by one and the same describing 
circle. 

Let A and B be the two pitch circles. 

Take a circle of any convenient size less than either A or B, 
as indicated by the small dotted circle, and with it describe an 
epicycloid upon A and a hypocycloid upon B. 



Teeth of Wheels. 



175 



FIG. 192. 



Let these curves determine the acting surfaces aP, f>P, of two 
teeth in contact at P ; then the tooth aP will press against bP so 
that the perpendicular to the 
surfaces in contact at P shall 
pass through D, and the relative 
angular velocities of two pieces 
centred at A and B, and fur- 
nished with these teeth, will be 
the same as those of the two 
pitch circles. 

As the wheels rotate, we find 
that the point of contact P tra- 
vels along the upper small dotted 
circle starting from D. Tn the 
same way the points of contact of 
teeth to the left of ADB travel 
along the lower dotted circle up 
to D, and it is, therefore, essential 
to form the teeth in the manner 
which we are about to describe 
by somewhat extending our con- 
struction. We have now to make 
complete teeth upon both wheels, 
and to provide that either A or 
B may be the driver. 

As far as we have gone we 
have described the point of a 
tooth upon A and the flank of 
one upon B, and have supposed A to drive B. If the conditions 
were reversed, and B were to drive A, we should have to obtain 
from one describing circle the curves suitable for the point of a 
tooth upon B and the flank of one upon A. This describing circle 
is not necessarily of the same size as the former one, but it is very 
advantageous to make it so, and we shall therefore assume that the 
teeth upon A and B are formed by the same describing circle. 

Let the describing circle G trace PQ, SR upon A, and /^, sr 
upon B, then the complete teeth can be made up as shown in the 
diagram on the next page (see fig. 193). 




176 



Elements of Mechanism. 



By preserving a constant describing circle, any wheels of a set 
of more than two will work together, as, for example, in the case 
of change wheels in a lathe. 

FIG. 193. 




FIG. 194. 



It remains to discuss the character of the teeth as dependent 
upon changes in the form or configuration of the hypocycloidal 
portion of the curves. 

ART. 149. If we trace the changes in form of the hypocycloid, 
as the describing circle increases in size until its diameter be- 
comes equal to the radius of the circle in which it rolls, we shall 
find that the curve gradually opens out into a straight line. 

It is indeed a well-known geometrical fact that when the 
diameter of the circle which describes a 
hypocycloid is made equal to the radius 
of the circle within which it rolls, the 
curve becomes a straight line. 

Let C be the centre of the describing 
circle at any time, and let P be the corre- 
sponding position of the describing point 
(fig. 194). 

Suppose that P begins to move from 
E, so that the arc PQ shall be equal to 
the arc EQ. 

Join CP, AE; let EAQ = 0, PCQ = </>. 
ThenPQ = arcEQ, or 




Teeth of Wheels. 



177 



But AE = 2 CQ, /. CQ x <f> = zCQ x 8 or = 28. 

Now cannot be equal to 26 unless P coincides with R in 
the line AE, in which case the diameter EAD is the path of P. 

This property of a hypocycloid is taken advantage of in 
Wheatstone's photometer, where an annular wheel is constructed, 
and a second wheel of half its diameter is made to run very 
rapidly upon the internal circumference : a small bead of glass, 
silvered inside, is attached to a piece of cork fitted on this internal 
wheel The bead would give the images of two lights held upon 
either side of it. When the wheel revolves these small images or 
spots of light become luminous lines of light, whose brilliancy can 
be compared, and made equal, by shifting the apparatus towards 
the weaker light This contrivance is a philosophical toy, it is 
not used. 

ART. 150. The first particular case of the general solution is 
the subject of the present article. 

It will be remembered that the hypocycloid determines the 
flank of the tooth upon either wheel : if, therefore, the radius of 
the circle describing the hypocycloid be taken in each case to be 
half that of the corresponding pitch circle, the teeth will have 
straight, or radial flanks, as they are 
commonly called. 

The method of setting out the 
teeth is the following : 

Let A and B be the centres of 
two pitch circles which touch in the 
point D. 

Let a circle, F, whose diameter is 
equal to BD, roll upon the circle A, 
and generate the epicycloid QP : 
this curve determines the form of 
the driving surface of the teeth to 
be placed upon A. 

Let another circle, G, whose dia- 
meter is equal to AD, roll upon the 
circle B, and generate the epicycloid 
qp : this curve determines the driv- 
ing surface of the teeth to be placed upon B. 



FIG. 




178 



Elements of Mechanism. 



Here of necessity the describing circle is not of the same size 
FlG 196 when tracing out the points of 

the teeth upon A and B ; but, 
by reason that the same circle 
gives the point upon A and 
the flank upon B, or con- 
versely, and that the flanks 
in each case are straight 
lines, the condition in Art. 
146 is still fulfilled. 

The annexed figure shows 
us these teeth with radial 
flanks, the straight edges of 
the teeth pointing towards 
the centres of the respective 
pitch circles. 

ART. 151. As the circle 
describing the hypocycloid 
goes on increasing until it 
becomes equal to the circle 

in which it rolls, the curve passes from a straight line into a curve, 
and finally degenerates from a small half-loop shown in the sketch 
down to an actual point. 

FIG. 197. 






It appears also that the same hypocycloid is generated by 
each of the circles A and B, which are so related that the sum 
of their diameters is equal to the diameter of the circle in which 
^hey roll. 

ART. 152. The second particular case of the general solution 
occurs when the hypocycloid degenerates to a point : we then 



Teeth of Wheels. 



179 



FIG. 198. 




obtain a wheel with pins in the place of teeth, and derive a form 
which is extensively used in clockwork. 

There is a very old form of pin wheel, called a lantern 
pinion, where the pins are made of round 
and hard steel wire, and are supported be- 
tween two plates, in the manner shown 
in the sketch. This form has been much 
used by clockmakers, because it runs 
smoothly, and has the merit of combining 
great strength with durability. 

The pin must have some sensible diameter, but we will first 
suppose it to be a mathematical point. 

We have just seen that when the 
hypocycloid becomes a point, the 
describing circle must be taken equal 
to that within which it is supposed 
to roll. 

As before, let A and B be the 
centres of two given pitch circles 
which touch each other in the 
point D. 

Let a circle, F, equal to B, roll 
upon the circle A, and generate the 
epicycloid PQ. 

This curve will determine the 
acting surface of the teeth to be 
placed upon A, which will work 
against pins to be placed at equal 
intervals on the circumference of 
the circle B. 

Thus we shall have epicycloidal 
teeth upon the driver, working with 
hypocycloidal teeth on the follower, 

but these latter teeth are pins, or mere points theoretically, instead 
of being curved pieces of definite form. Here it is perfectly ap- 
parent that the condition upon which we rely is again fulfilled. 

The pin must have some size, and we shall take into account 
the size of the pin by supposing a small circle, equal to it in 

N 2 




i8o 



Elements of Mechanism. 



sectional area, to travel along the theoretical path of the point, 
and to remove a corresponding 
portion of the curved area occu- 
pied by the epicycloids. 

Assume that QP represents 
the acting surface of a tooth 
which drives before it a point, P 
(fig. 200). 

Make P the centre of a circle 
equal to the size of the pin: 
suppose this circle to travel along 
PQ, having its centre always in 
the curve : remove as much of 
the tooth as the circle intercepts, 
and the remainder will give the 
form of the working portion. 

We shall presently find that 
in practice the pins are always 
placed upon the driven w/iect, and as this rule is never broken, 
for reasons to be stated hereafter, we shall assume it to exist when 
we come to apply our solution to the case of a rack and pinion. 

ART. 153. If either of the wheels becomes a rack, that is, 
straightens into the form of a bar, the radius of the pitch circle 
must be infinitely large ; and we shall now take up the inquiry as 
to the changes introduced into the shape of the teeth by this 
transition from the circle into a straight line. 





The curve which we have called an epicycloid changes into a 
cycloid when the rolling circle runs along a straight line instead of 
upon the outer circumference of another circle, 



Teeth of Wheels. 181 

It is, in fact, the curve described by a point in the rim of a 
wheel as it runs along a level road or rail. 

It is shown in fig. 201, and possesses some very interesting 
properties with reference to the swing of a pendulum ; it is, there- 
fore, a curve very familiar to those who study mechanics. 

So far as the general solution in Art. 148 is concerned, the 
changes will be the following. Conceive that the circle A is 
enlarged till it becomes a straight line ; then the circle G, which 
rolls upon the inner and outer circumferences of the circle A, 
tracing thereby the points and flanks of the teeth upon A, will in 
each case generate the same curve, viz., a cycloid. 

Thus the teeth upon B will remain as before, and each face of 
a tooth upon the rack A will be made up of two arcs of cycloids 
meeting in the pitch line. 

ART. 154. In Art. 150, where the teeth have radial flanks, 
the matter is not quite so simple, for the describing circles which 
give the radial flanks are in each case to be of one-half the dia- 
meter of the pitch circle in which they respectively roll ; and here 
one of the pitch circles is infinite, 
whence it follows that a circle half its 
diameter is infinite also, or may be re- 
garded as a straight line. 

The curve traced out by one ex- 
tremity of a straight line rocking upon 
the circumference of a given circle, is, 
of course, the same as that described 
by one end of a string PQ, which is 
kept stretched while it is unwound from 
the circumference of the circle. The 
end of the line, or the end of the 
string, is at first at the point A in the curve AP, and the curve is 
traced out while the line rocks in one direction, or during the un- 
winding of the string. 

This curve AP is a very well-known curve, and is called the 
involute of a circle. We have met with it before, and we proceed 
to show that in the case of a rack and pinion having teeth with 
radial flanks, the driving surfaces of the teeth upon the pinion will 
be the involutes of the pitch circle of the pinion in question. 




182 



Elements of Mechanism. 




ART. 155. To make this matter clear, we refer to fig. 203, 
and observe that the circle F, 
rolling upon a straight line, 
generates a cycloid and gives 
the form of the driving sur- 
faces of the teeth upon the 
rack: the circle G becomes 
infinite, and E/ changes to a 
straight line. The change 
which the rest of the con- 
struction undergoes is simply 
the substitution of the invo- 
lute qp for the corresponding epicycloid, the circle G having 

passed into a straight line. 
The change is scarcely 
visible to the eye, but the 
form of the teeth is shown 
in the diagram, where the 
curved portions in the rack 
are cycloids, the radius of 
the describing circle being 
half that of the pitch circle 
of the pinion, and the curves upon the pinion are the involutes of 
its own pitch circle. 

ART. 156. Where pins are substituted for teeth in either the 
rack or the pinion, we construct in accordance with the rule that 
the //>/.$ are always placed upon the follower. 

FIG. 205. FIG. 206. 



FIG. 204. 






i. Let the rack drive the pinion. 

Here the circle A becomes infinite, and the curve PQ passes 



Teeth of Wheels. 



183 



into a cycloid, so that the teeth upon the rack are cycloidal, as 
shown in fig. 205. 

2. Let the pinion drive the rack. 

Here the circle B becomes infinitely large, and CP changes 
into a straight line, the curve PQ passing into the involute of a 
circle, with the result exhibited in fig. 206, where the teeth of the 
driver are the involutes of a circle and are known as involute teeth. 

ART. 157. The last case to be brought before the reader is 
derived from a property of this involute of a circle, and the teeth 
are very easily obtained, but are not used in practice, on account 
of their being unsuited for the transmission of any considerable 
forces. 

We proceed to show that the geometrical requirements of our 
construction are fulfilled completely by involute curves. 

Let A and B represent the centres of two pitch circles touch- 
ing at the point D, as shown by the dotted lines, and with B as a 
centre, and any line BQ, less than BD, as radius, describe another 
circle. Through D draw DQ touching this smaller circle, draw 
AR perpendicular to QD produced, and with centre A and radius 
AR describe a circle touching QR 
in the point R. 

If now we take any point P in 
QR, and describe the involutes EP 
and FP by winding two portions of 
strings, such as PQ and PR, back 
again upon their respective circles, 
we shall have two forms of imaginary 
teeth in contact, viz., EP and FP, 
such that 

(1) These teeth have a common 
perpendicular to their surfaces at P, 
viz., RPQ. 

(2) This perpendicular cuts the 
line of centres in a fixed point D. 

But these are the conditions 
which we are seeking to fulfil. 

No more direct illustration of our leading proposition could 
be conceived than this one. 




184 



Elements of Mechanism. 



The lines PR and PQ are the respective radii of curvature 
of the involute curves in contact at P, while RQ, which is equal 
to RP + PQ, is the link of constant length connecting the arms 
AR and BQ. 

The angular velocities of AR and BQ are therefore as BQ to 
AR, or as BD to AD, and this ratio remains constant so long as 
the curves EP, FP remain in contact. 

ART. 158. In order to construct the teeth we must draw our 
FlG 2og pitch circles touching in 

D, and then select some 
angle BDQ at which to 
draw the line RDQ. 
When this angle is de- 
termined, we obtain the 
circles of radii BQ, AR, 
by dropping perpendicu- 
lars upon RDQ from the 
centres A and B, and we 
then describe teeth of 
the required pitch by 
constructing the invo- 
lutes of these two cir- 
cles respectively. 

We observe, of course, 
that a great latitude is 
introduced from the circumstance that AD and BD remain con- 
stant while AR and BQ may have different values. 

In teeth of this kind there is no difference between the point 
and the flank : the whole of each edge of a tooth is one and the 
same curve, viz., the involute of one of the two arbitrary 
circles. 

And further, the points of contact of two teeth must lie either 
in the line RDQ, or in a second line passing through D, and 
touching both the circles upon the opposite sides. 

ART. [59. To adapt this solution to the case of a rack and 
pinion, we note that one of the circles becomes infinite, and, fur- 
ther, that the involute of the infinite circle of radius AR is a 
Straight line perpendicular to its circumference, or perpendicular 




Teeth of Wheels. 



185 




to QD. Hence the teeth of the rack are straight lines perpendi- 
cular to the direction of 
QD. 

The direction of DQ 
is arbitrary; but when it 
has once been assumed, 
the radius BQ will be 
determined, and involute 
teeth can be formed upon 
B, the teeth of the rack 
being straight lines in- 
clined to the pitch line at an angle equal to BDQ. 

ART. 160. There are now sundry general points for consi- 
deration. We may inquire, where does FIG 
the action of two teeth begin, and 
where does it leave off? 

Referring to the solution in Art. 
148, we observe that if the motion 
takes place in the direction of the 
arrows, and the describing circle be 
placed so as to touch either pitch circle 
in D, the contact of two teeth com- 
mences somewhere in #D, travels along 
the arc aDl>, and ceases somewhere in 
D& 

Since aD lies entirely without the 
pitch circle B, it is clear that the action 
in aD is due solely to the fact that 
the teeth upon B project beyond the 
pitch circle B, and similarly that the 
action in D& depends upon the projections or points of the teeth 
upon A. 

It is further evident that the greater the number of teeth upon 
the wheels, the closer is their resemblance to the original pitch 
circles, and the more nearly is their action confined to the neigh- 
bourhood of the point D. 

By properly adjusting the amount to which the teeth are 
allowed to proiect beyond the pitch circles, and also their num- 




1 86 Elements of Mechanism. 

bers, we can assign any given proportion between the arcs of con- 
tact of the teeth upon either side of the line ADB. 

Where the teeth upon B are pins, there is comparatively very 
little action before the line of centres, and there would be none at 
all if the pins could be reduced to mere points, as in that case 
there would be nothing projecting beyond the pitch circle B. 

Again, since the line DP in Art. 147 is a perpendicular to the 
surfaces in contact at P, it follows that the more nearly DP re- 
mains perpendicular to ADB, the less will be the loss of the force 
transmitted between the wheels. 

Here we have an additional reason for keeping the arc of con- 
tact as close as possible to the point D. There is a sensible loss of 
power as soon as the line DP differs appreciably from the direc- 
tion perpendicular to AB. 

It is on this account that involute teeth are not used in ma- 
chinery calculated to transmit great force. The line RPDQ in 
Art. 157 is always inclined to the line ADB at a sensible angle, 
and a direct and useless strain upon the bearings of the wheels is 
the result. 

ART. 1 6 1. In combinations of wheel work, the accurate po- 
sition of the centres must be strictly preserved. All the solutions 
given above, with one exception, entirely fail if there be any error 
in centring the wheels ; they are totally vitiated if anything arises 
to deprive them of their geometrical accuracy. The exception 
occurs with involute teeth : the position of the centres determines 
the sum of the radii of the pitch circles, and the wheels will work 
accurately as long as the teeth are in contact at all. 

We see too that teeth with radial flanks are not suitable for a 
set of change wheels ; the describing circles of one pair of wheels 
are derived directly from their pitch circles, and cannot be adapted 
to any other pair in the series. 

Where, however, the solution in Art. 148 is employed, the 
describing circles may be made the same for all the pitch circles,, 
instead of varying with each one of the series, and in that case 
any pair of wheels will work truly together. 

As regards the strength of the teeth, we remark that this quality 
is influenced by the size of the describing circle. 

If the diameter of the describing circle be less than, equal to, 



Teeth of Wheels. 187 

or greater than the radius of the pitch circle, we shall have the 
flanks as shown in the sections a, b, c of the sketch. 




It is evident that a small describing circle makes the teeth 
strong, and that it would be unwise to have them weaker than 
they are with radial flanks. The form of involute teeth being 
somewhat similar to that of a wedge, the teeth of this character 
are usually abundantly strong. 

It will be proved, when we treat of rolling curves, that the 
surface of one tooth must always slide upon that of another in 
contact with it, except at the moment when the point of contact 
is passing the line of centres. 

This matter should be well understood, the teeth are per- 
petually rubbing and grinding against each other ; we cannot pre- 
vent their doing so : our rules only enable us so to shape the 
acting surfaces that the pitch circles shall roll upon each other. 

Nothing has been said about the teeth rolling upon each other; 
it is the pitch circles that roll ; the teeth themselves slide and rub 
during every part of the action which takes place out of the line 
of centres. 

Since, then, the friction of the teeth is unavoidable, it only re- 
mains to reduce it as much as possible, which will be effected by 
keeping the arc of action of two teeth within reasonable limits. 

Generally, the friction before a tooth passes the line of centres 
is more injurious than that which occurs after the tooth has passed 
the same line : the difference between pushing a walking-stick 



188 



Elements of Mechanism. 



along the ground before you and drawing it after you has been 
given as an illustration of the difference between the friction 
before and after the line of centres ; but this difference is less 
appreciable when the arc of contact is not excessive. 

Where a wheel drives another furnished with pins instead of 
teeth, the friction nearly all occurs after the line of centres ; 
hence such pin wheels are very suitable for the pinions in clock- 
work. 

ART. 162. When the axes are not parallel we must employ 
bevel wheels, the teeth upon which are formed by a method due to 
Tredgold. 

FIG. 212. 




Let FEDH, KEDL, represent the frusta of two right cones, 
whose axes meet in C, and which are therefore capable of rolling 
upon each other. 

Let it be required to construct teeth upon two bevel wheels 
which shall move each other just as these cones move by rolling 
contact. 



Teeth of Wheels. 189 

Draw ADB perpendicular to DE, meeting the axes of the 
cones in the points A and B. 

Suppose the conical surfaces, HAD, BDL, to have a real exist- 
ence, and to be flattened out into the circular segments DR, DS : 
these segments will roll upon each other just as the circular base 
HD rolls upon the circular base DL. 

Hence these segments will serve as pitch circles, upon which 
teeth may be constructed by the previous rules : such teeth may 
be formed upon a thin strip of metal, and their outline can then 
be traced upon the surface of the cone terminating in A. 

Similarly, if ^Ea be drawn perpendicular to ED, the circle of 
radius Ad = Ea will be the pitch circle for the teeth upon the 
conical surface EaF. The teeth will taper from D to E, and the 
intermediate form will be determined by a straight line moving 
parallel to itself, and originally passing through the points D and E. 

It is stated in Buchanan's account of this method that ' the 
length of the teeth, the friction of them, and the peculiar advan- 
tages of the different modes of forming them, may be considered 
on the developed pitch lines in the same manner as if they were 
the pitch lines of spur wheels ; consequently every remark that 
applies to the one, applies to the other. Indeed, the only difficulty 
in this construction of the teeth of bevelled wheels consists in 
applying the patterns correctly to the conic surface whereon the 
ends of the teeth are to be described.' 



90 



Elements of Mechanism. 



CHAPTER VI. 

ON THE USE OF WHEELS IN TRAINS. 

ART. 163. When a train of wheels is employed in mechanism, 
the usual arrangement is to fasten two wheels of unequal size upon 
every axis except the first and last, and to make the larger wheel 
of any pair gear with the next smaller one in the series. 

FIG. 213. 




Let A be the driver, L the extreme follower, and conceive that 
L makes (e) revolutions while A makes one revolution ; 

number of revolutions of L in a given time 

"number of revolutions of A in the same time ' 
It will be convenient to distinguish (e) as the value of the train, 
and the ratio which it represents may be at once found when the 
numbers of teeth upon the respective wheels are ascertained. 

Suppose that A, B, C, D, &c., represent the numbers of teeth 
upon the respective wheels, thus we infer from the condition of 
rolling that 

number of revolutions of B in a given time _ A 
number of revolutions of A in the same time~B ' 
and similarly for each pair of wheels : 

ACE K 



Wheels in Trains. 191 

It may frequently simplify our results if we regard e as positive 
or negative according as A and L revolve in the same or in oppo- 
site directions : thus, in a train of two axes, e would be negative, 
and in a train of three axes it would be positive. 

The comparative rotation of wheels is estimated in various 
ways, thus : 

Let N, n be the numbers of teeth upon two wheels, such as 

A and B. 
R, r their radii. 

P, / their periods of revolution. 
X, x the number of revolutions made by each wheel in 

the same given time. 
It is easy to see that 



Note. A belt and a pair of pulleys supply a mechanical 
equivalent for spur wheels : the belt may be open or crossed, and 
in either case 

the number of revolutions of B in a given time 
the number of revolutions of A in the same time 
_ diameter of A 
"diameter of B ' 

The crossing of the belt merely reverses the direction of one of 
the pulleys. Whence it follows that two pulleys with a crossed 
strap are equivalent to two spur wheels in gear, but that if the 
strap be open the combination is equivalent to three spur wheels. 
Ex. Suppose that we have a train of five axes, and that 

1. A wheel of 96 drives a pinion of 8. 

2. The second axis makes a revolution in 1 2 seconds, and the 
third axis in 5 seconds. 

3. The third axis drives the fourth by a belt and a pair of 
pulleys of radii 20 and 6 inches. 

4. The fourth axis goes round twice while the fifth goes round 
three times. 

96 12 20 3 
Here < = * 8 -x--x y x*= 144, 

or the last axis makes 144 revolutions while the first axis goes 
round once. 



192 



Elements of Mechanism. 



ART. 164. An example of the communication of motion direct 
from the fly-wheel of an engine to a rotating fan by means of 
pulleys and bands is given in one of Sir J. Anderson's diagrams. 

Here the beam of the engine vibrates through the arc a&, and 
the crank pin at the end of the connecting rod describes a circle, 
the diameter of which is to that of the fly-wheel A as 4 to 12, or 
as i to 3. Let the mean pressure on the crank pin = 6000 Ibs., 
then the tangential pressure at circumference of fly-wheel is equal 




to 2000 Ibs. It will be seen that the motion is carried from the 
fly-wheel A to the pulley B by an open strap, then from the pulley 
C to D by means of a crossed strap, then from E by an open strap 
to F, the fan. In each case there is an increase of the speed of 
rotation of the driven pulleys, and a corresponding decrease in the 
driving pressure. 

According to the numbers set out on the diagram the fly-wheel 
makes 20 revolutions per minute, and the fan makes 1600 revolu- 
tions in the same time, the rate of increase being arrived at by a 
comparison of the respective diameters of the drivers and followers. 

In like manner the tension of each band may be deduced 



Wheels in Trains. 



193 



from the principle of work, by observing that the product of the 
tension of a band into its linear velocity is constant. Whence, 
linear velocity of band on C I linear velocity of band on B ; ; 8 I 3. 

Therefore, tension of band on C=^ * 2000 Ibs. = 750 Ibs. 
In like manner, tension 8f band on E=- x 750 Ibs. =300 Ibs. 
The results are tabulated on the diagram as follows : 





Diameter 


Revolutions 


Pressure 


A 


12 


2O 


2OOO 


B 


3 


80 


2COO 


C 


8 


80 


750 


D 


2 


320 


750 


E 


5 


3 20 


300 


F 


i 


I6OO 


300 



ART. 165. It is very obvious that a wheel and pinion upon 
the same axis is a combination equivalent to a lever with unequal 
arms, and modifies the force which may be transmitted through 
it, and, further, that a single wheel is equivalent to a lever with 
equal arms, and produces no modification in the force which may 
pass througrrtt. 

So, therefore, when any number of separate wheels are in gear, 
no two wheels being upon the same axis, they are equivalent to a 
single pair of wheels, viz., the first A, and the last L : the inter- 
mediate wheels act as carriers only, and transfer the motion 
through the intervening space. 

FIG. 215. 




This also appears from the formula, where we find that 



ABC 



K A 



: B X C X D X ' L~L' 



194 Elements of Mechanism. 

which is the same result as if A and L were alone concerned in 
the movement. 

>^&RT. 1 66. If, however, a single wheel, such as B, be inter- 
posed between two other wheels A and L, although B will not 
modify the force transmitted, nor alter the velocity, it may be 
useful in changing the direction in which the wheel L would 
otherwise revolve. An intermediate wheel so introduced is tech- 
nically called an idle wheel, and we give instances where this 
intermediate wheel serves a very useful purpose in causing twc 
other wheels to rotate in the same direction with precisely the 
same velocity. 

1. The student will remember the peculiar heart-shaped cam 
for driving the needle bar in a sewing machine, as well as the 
combination of two cranks and a link for giving motion to the 
shuttle, and he will find on looking back that in each case the 
driver was a pin rotating in a circle. 

The machine from which these movements were taken illus- 
trates a combination of three spur wheels in 
gear, the central wheel being the driver, and 
the other two wheels being equal and re- 
volving in the same direction with equal ve- 
locities. 

There is a small fly-wheel driven by hand, 
f / \ on the axis of which is the spur wheel B, and 

11 I the object being to cause two parallel axes at 

' A and C to rotate in the same direction, and 
at the same rate, it is arranged that equal 
f ( ^ \ spur wheels, A and C, shall both gear with B, 

vv ) in the manner shown by the pitch circles 

marked in the sketch. 

2. The Blanchard turning-lathe, of which a portion is shown 
in the sketch, is used for shaping the spokes of wheels, gun-stocks, 
shoe-lasts, and other pieces of an arbitrary form, which no one 
could imagine, until the method was explained, as being the sort 
of objects that would probably be turned in a lathe. 

But the solution is, that this copying principle admits of end- 
less application, and it will be seen that if we place two lathes side 
by side, and cause the actual cutter in the one to copy exactly the 




WJieels in Trains, 



195 



form which an imaginary cutter is tracing out upon a model in the 
other, we shall reproduce upon a piece of wood placed in the 
second lathe the precise pattern which exists as the copy. 

In the drawing the mandrels of the two lathes are shown at F 
and G, the dark oval at F representing a section of the spoke of a 
wheel, and being, in fact, an exact copy in iron of the thing to be 
manufactured. The spoke F is attached to the wheel A, while B is 
an intermediate wheel 'or driver, and C is another wheel of the 
same size as A. 

FIG. 217. 




The unfinished spoke is placed parallel to the copy, along the 
axis of the wheel C, and the function of the intermediate wheel, 
or driver B, is to cause the material to revolve in the same direc- 
tion and at the same rate as the pattern. 

A sliding frame, K, carries a tracing wheel, I, with a blunt 
edge, which is kept pressed against the pattern by a weight or 
spring, and also contains the cutters, H, which are driven at a 
speed of about 2,000 revolutions per minute by an independent 
strap. 

The circle described by the extremities of the cutters is pre- 
cisely the same size as the circle of the tracer, and it follows that 
the exact form which the tracer feels, as it were, upon the pat- 
tern, will be reproduced by the whirling of the cutters against the 
material, G, and that the spoke may be completed by giving a 
slow motion to the combination in a direction parallel to the axis 
of the pattern. 

Sometimes the tracer and cutters are mounted upon a rocking 
o 2 



196 



Elements of Mechanism. 




FIG. 219. 



frame, instead of upon a slide rest, but the principle of the ma- 
chine is not changed thereby. 

An ; ntermediate wheel may also be 
useful when two parallel axes are so close 
together that there is not space for the 
ordinary spur wheels. 

In such a case the axes A and C may 
be connected by a third wheel, B, and will 
of course revolve in the same direction. 

The wheel, B, is elongated so as to 
gear with both A and C, and is called a 
Marlborough wheel. The axes might also 
be connected without wheelwork, as we shall see hereafter. 

ART. 167. Speed pulleys are so called because they allow of 
the, transfer of different velocities of rotation from one shaft to 
another : they are much used in 
engineers' factories. 

They are made in a series of 
steps, as shown in the diagram, one 
pulley being the counterpart of the 
other, but pointing in the opposite 
direction. 

If the steps be equal, as is com- 
monly the case, the sum of the radii 
of each pair of opposite pulleys will 
be a constant quantity. 

It is a geometrical fact that 
when two circles are placed with 
their centres at a given distance, 
and are so related that the sum of 
their radii remains constant, an end- 
less crossed band connecting both the circles will not vary in 
length in the smallest degree during the change in the actual dia- 
meter of each circle. 

Hence a crossed strap will fit any pair of the pulleys in our 
series with perfect exactness. 
The proof is the following : 
Let A and B represent two pulleys whose radii a.re AP and 




Speed Pulleys. 



197 



BQ, and assume that AP + BQ remains unchanged while AP and 
BQ respectively increase and diminish, 

and let AP = *, BQ=^, PAC = DBQ = 0. 
Then CPQD = xQ t yd + PQ, 



But it is clear that if AP be increased by a given quantity, and 
BQ be diminished by the same quantity, we shall not change the 
length of PQ, by reason that the alteration will only cause PQ to 
move through a small space parallel to itself between the lines 
AP, BQ, which are also parallel. 

Also x +y is constant by hypothesis, 

/. CPQD remains unaltered in length so long as our condition 
holds good. 

It may be interesting to examine this matter a little further, 
and to find an expression for the length of an open band connect- 
ing t\vo given pulleys. We will assume that we are dealing with 
step pulleys, the sum of the radii being za in every case, and x 
being the depth of the step or steps, or the quantity by which 
either radius differs from the assumed 
value of the semi-sum of the radii FlG- 220- 

Let AP=a+x, 



PAa = = QC/>, 
and let /= length of the band PQRS. 

Then the curved portions of the band 
resting upon the pulleys are (T + 23) (a + x) 
and (TT 20) (a x) respectively. 



+ (rr-28)(-.x 



Now --^4 = cos 0, 
AC 

.'. PQ = AC cos = f cos 0, 




,*. 2X = AC sin = c sin 0, 
. /= 2ira + 2cd sin + 2^ cos d. 



198 Elements of Mechanism. 

It is evident that / is no longer constant, and that it must neces- 
sarily change when x or changes: still the variation of length 
may be so little as to be disregarded under the ordinary propor- 
tions occurring in a workshop. 

Since would seldom represent an angle so large as 10, and 
we have pointed out in Art 1 1 1 how small a difference exists be- 
tween sin 8 and within even larger limits, we will assume that 
sin = 0, 

then cos 0=1-2 sin 2 ^= i- ^ = i- 6 *. 

2 4 2 

/ / fl2 \ 

Therefore /= 2*0, + 2OT-f 2C (\ -- J, 



Call /' the value of / when x=o, or = o 



But 2X = 



which expresses the difference of the lengths in a convenient 
form. 

It is apparent at once that / is greater than /'. 

It has been stated that this difference is very trifling in many 
cases, and the following example is an illustration. 

Let the diameters of the steps of the pulleys be 4, 6, 8, 10, 12 
inches respectively, and let / be the length of strap for the pair 
of 12 and 4, while /' is the length for the equal pair of 8 and 8, 
the distance between the centres of the pulleys being 6 feet. 

Then /-/'= 4 (^4) 2 =i^=2 inch whkh is rather Jes3 than 

72 72 9 
of an inch. 

In practice, open bands are usually preferred to those which 
are crossed. The latter embrace a larger portion of the circum- 
ference, and are therefore less liable to slip, but they rub and wear 
away at the point where they cross. 



Speed Pulleys. 



199 



ART. 1 68. As an example of the use of speed pulleys, we 
refer to the contrivance sketched in fig. 221, which is to be found 
in every large lathe, and is useful in other machinery, where it is 
required to obtain increased power or a diminished speed. It 
enables the mechanic to change the velocity of the mandrel of the 
lathe, and gives another simple example of the use of wheels in 
trains. 

There is a driving shaft overhead, provided with a cone pulley, 
and with fast and loose pulleys, which receive the power from 
the engine : a second 
cone pulley, F, is fitted 
on the spindle of the 
lathe, and rides loose 
upon it : to this cone is 
attached a pinion G, 
which drives a wheel H, 
and so the motion is 
communicated by the 
pinion K to the wheel 
L, which is fastened to 
the mandrel of the lathe, 
and turns with it. The 
result is that the wheel 
L revolves much more 
slowly than the cone 
pulley F, and that the 
speed of the mandrel is 
reduced by the multi- 
plier ** where G, 
xl X Li 

K, H, L represent the 
numbers of teeth upon 
these wheels respec- 
tively. 

Where the lathe is 
worked at ordinary 
speeds, the wheels H 
and K are pushed out of gear by sliding the piece HK in the 




2OO 



Elements of Mechanism. 



direction of its axis, as shown in the lower diagram, and the cone 
pulley, F, is fastened to L by a pin. 

This pin must of course be removed as soon as the slow mo- 
tion comes into work. 

As this movement is very similar to the gearing in a crane, we 
shall presently examine the application of these trains in raising 
heavy weights, and shall see how they may be applied so as to 
reduce velocity, and thereby to increase the amount of force 
which is called into play. 

After what has been stated it is scarcely necessary to point 
out the express use of conical pulleys : they form an obvious 
modification of step pulleys where the change is continuous in- 
stead of being abrupt. 

There are two forms, one where the oblique edges of a section 
are parallel straight lines, and the other where the convexity of 
one section exactly fits into the concavity of the other. 

If the band be crossed we 
have seen that it will retain the 
same tension in every position 
upon the cones. If it be open, 
it will be less stretched at the 
middle than at either end, ac- 
cording to Art. 167. When the 
obliquity is small, the difference 
becomes absorbed in the elas- 
ticity or ' sag ' of the band ; 
otherwise it must be provided for by giving convexity to one or 
both of the cones. 

The rotation of the upper cone being uniform, it is evident 
that the rotation of the lower cone will decrease as the strap is 
shifted towards the right hand. 

One of the cones is sometimes replaced by a cylindrical drum, 
in which case the strap must be kept stretched by a tightening 
pulley. 

As an illustration, we refer to the use of these conical pulleys 
in the manufacture of stoneware jars and other large earthenware 
vessels, where a mass of clay is fashioned into the required form 
upon a rotating table, and the workman varies the speed of the 




Clock- Train. 



201 



table according to the requirements of the work by shifting the 
driving strap along a pair of cones. 

ART. 169. A common eight-day clock affords a familiar illus- 
atioh of the employment of a train of wheels. 
We have marked the disposition of the wheelwork in a clock 
of this character, and the various wheels are named in the sketch. 

The great wheel turns round once in 12 hours, and may have 
96 teeth. Suppose it to engage with a pinion of 8 teeth on the 
axis or arbor of the centre wheel, this pinion will turn twelve 
times while the great wheel turns once, and is capable of carrying 
the minute hand. Let the pendulum swing 60 times in a minute, 
or be a seconds' pendulum, the scape wheel will then have 30 
teeth, and will be required to turn once in a minute. 

Hence the value of the train from the centre to the scape 
wheel should be 60 ; and in constructing the train we observe 
that if the pinions on the axes of the second and scape wheels 
have each of them 8 teeth, 



the centre and second 
wheels may have 64 and 
60 teeth. In such a case 
we should have 

^ 6 8x8 0==6 - 

In order that the clock 
may go for 8 days, the 
great wheel must be cap- 
able of turning 16 times 
before the maintaining 
power is exhausted. 

It is easy to see that 
if the speed of the scape 
wheel at one end of the 
train be increased, and if 
we are at the same time 
limited in respect of the 
number of rotations of the 
great wheel, it will be con- 
venient to introduce a new axis into the train 



FIG. 223. 




and, accordingly, 



202 Elements of Mechanism. 

an additional wheel and pinion is found in the train of a watch, 
where the balance wheel, which performs the function of a pen- 
dulum, makes at least 120 vibrations in a minute. 

Another illustration of a train of wheels is found in the 
method of driving the hour hand of a clock or watch, and in 
order to understand it we have only to observe that in a clock or 
watch the minute hand is fastened to the arbor or axis of the 
centre wheel, and that the hour hand is attached to a pipe which 
fits upon this axis, and derives its motion from the minute hand. 

This appears from the diagram, and all we have to do is to 
connect the pipe and axis by a train of wheels which shall reduce 
the velocity in the ratio of i to 1 2. 

FIG. 224, 
HOUR HAND H- 




The drawing is taken from a small clock, and represents the 
train of wheels employed. The pinion K, attached to the axis of 
the minute hand, drives H, whence the motion passes through G 
to L, and thus to the hour hand, which is fastened to the pipe on 
which L is fitted, and which corresponds to the mandrel of the 
lathe. The value of e in the train is given by the equation 
_KxG_28x 8 , 
~H x L~ 4 2 x 6 4 ~~ Tir 

ART. 170. The mechanism of a lifting crab for raising 
weights affords an elementary example of the use of a train of 
wheels. 

The diagram is from Sir J. Anderson's series. On the right 
hand there is an elevation of the crab, showing the upper shaft, 
AB, to which the driving lever handles are attached. 

The radius of the circle described by the extremity of the 
lever handle is to that of the spur wheel C as 5 : i. 



Lifting' Crab. 203 

Again, the radius of the spur wheel D is to that of the drum 
as 5 : i. 

Hence the power is to the weight raised as 25 : i. 

Let P=6o Ibs., then resistance overcome by a rope wound 
round the drum=6ox 25 Ibs. = i5oo Ibs. 



FIG. 225. 



Mo 




Mo. 



The diagram is analysed for the use of a teacher. The mark- 
ings 'Dis. i,' 'Dis. 5,' &c., indicate the relative distances at which 
the forces act, and the markings ' Mo. 5,' ' Mo. 25,' give the key 
to the mechanical advantage gained. Thus, when the rope at- 
tached to the weight is pulled in by one foot, the corresponding 
motion of a point in the circumference of the large pitch circle 
on the axis of the drum is 5 feet, while the motion of the end of 
the driving handle is 25 feet. Hence, by the principle of work, 
60 x 2$=x x i, where x is the resistance overcome, 
therefore x=i$oo Ibs. 

The arrangement of wheelwork in a crane for raising the 
heaviest weights would be something of the character shown in 
fig. 226, with this difference, that the wheels would be broader 
and more massive as we approached the axis on which the weight 
directly acts. 



2O4 



Elements of Mechanism. 



We take a case in which four men, each exerting a force of 
15 ibs., could raise a weight of somewhat more than 4 tons. 

As we are only examining the theoretical power of the combi- 
nation, we will neglect the loss of power by friction. 

The men act upon the winch-handles, and the lengths of the 
arms of these handles are shown as being equal to the diameter 
of the drum on which the rope or chain is coiled. This gives a 

leverage of 2 to i. 

Fic - 22fi - We next observe that a large 

an d small wheel are placed upon 
^ each axis, and calling e the value 

Jj of the train, we have the relation, 

20 40 20 
e= x 2 x 

TOO I2O IOO 



Hence the wheelwork multiplies 
the power 75 times, while the 
proportion between the length 
of the winch -handles and the 
radius of the drum multiplies 
the power by 2, and thus we re- 
duce the velocity of the weight 
which is being lifted 150 times 
as compared with the rate at 
which the ends of the handles 
will move ; that is to say, the 
power exerted upon the weight 
is 150x60 Ibs., or 9000 Ibs., 

which is larger than 4 x 2240 Ibs., or larger than 4 tons expressed 

in pounds. 

ART. 171. It is a maxim among mechanics that all screws 

which are required to be perfectly accurate must be cut in a 

lathe, and there is a geometrical reason for this statement, de- 

pending upon the varying inclination of the screw surface at dif- 

ferent distances from its axis. 

In cutting a screw-thread upon a bolt without using a lathe, 

we employ pieces of a nut which would exactly fit the screw 




Screw Cutting, 205 

when finished, in order to carve out the thread. These pieces, 
which are called dies, are made of soft steel in the first instance, 
but are afterwards hardened and tempered, and have cutting edges. 
They are pushed forward by wedges towards the axis of the bolt, 
during the operation of cutting the thread. 

It follows that the angle of a ridge upon the die or cutter 
begins to trace out the screw-thread upon the bolt. But this 
angle corresponds to the inside line, in the hollow between two 
ridges, when the screw is completed. We begin, therefore, by 
tracing out a line, which is slightly different in inclination from 
the line of the thread that we require. The inclination of the 
thread, when the cutter begins its work, is not theoretically the 
same as when it leaves off. The difference is scarcely appre 
ciable, or even recognisable, in small screws ; but it exists not- 
withstanding, and we encounter in screw-cutting a practical diffi- 
culty which has never been absolutely overcome. 

We can only avoid this difficulty by having recourse to the 
lathe. 

In order to make this statement more intelligible, we refer 
to the sketch where RPM, RQN represent two right-angled 
triangles concerned in the formation of a screw-thread of a given 
pitch. 

Let PM=QN, and conceive that the triangle RPM is wrapped 
round a right cylinder, the circumference of whose base is RM, 
then RP will form a screw-thread ,, 

r IG. 227. 

whose inclination is the angle 
PRM. In like manner RQN 
may be wrapped round a larger 
cylinder, the circumference of 
whose base is RN, in which case 
RQ will be the screw-thread 
lying at an inclination QRN. 

Thus, for screw-threads of the 

same pitch the inclination is less M w 

as the cylinder on which the thread is traced becomes greater. 

Bearing this in mind, let D ABCE represent a section of a d' 
which is to be employed to carve out a screw-thread on a cylinrs 
whose axis is HH. The cutting edges at C and A first c 



206 Elements of Mechanism. 

upon the cylinder, and they correspond to the angles of the 
thread marked c, a, respectively. They therefore begin by tracing 
out a thread whose inclination is greater than it should be, and 
it is manifest that this difficulty is incurable so long as we are 
operating with ordinary dies. 

ART. 172. The principle of construction of the screw-cutting 
lathe will be apparent from the sketch. 

Here the copying principle receives one of its most valuable 
applications. The maker of a lathe furnishes a screw, shaped 
with the greatest care and exactness, and places this screw in a 
line parallel to the bed of the lathe. 

The lathe now carries within itself a copy, which can be re- 
produced or varied at pleasure, for by means of it we can advance 
the cutter so as to carve out any screw that we may require. 

The screw-thread which forms the copy is traced upon the 
F IG . 228. axis CD, and has a definite 

A B pitch assigned to it by the 

I maker. 

This screw carries a 
nut, N, and, disregarding 
the actual construction, we 
" i will suppose that the nut, N, 
is furnished with a pointer, 
P, capable of tracing a screw-thread upon another axis, AB. 

Conceive, now, that AB and CD are connected by a train of 
wheels in such a manner that they can revolve with any required 
relative velocities. 

Upon each revolution of CD the nut advances through a space 
equal to the pitch of the screw. If AB also revolve at the same 
rate as CD, and in the same direction, the point P will describe 
upon AB a screw-thread exactly similar to that upon CD. If AB 
revolve more or less rapidly than CD, the pitch of the screw upon 
AB will be less or greater than that upon CD. 

A train which would conveniently connect the axes is shown 
in fig. 229. Here C is the axis of the leading screw, and A carries 
the bar which is to be the subject of the operation ; it is, in fact, 

e mandrel of the lathe. 
we Let E, F, K, H represent the numbers of the teeth upon the 



Screw-Cutting Lathe. 



20? 



wheels so distinguished, and let e be the value of the train, and 
suppose AB to make (tri) revolutions for (n) revolutions of CD, 



we have therefore 
ExK 



But e-. 



FxH 



pitch of screw on AB n 

pitch of screw on CD m 
. pitch of screw upon AB_E x K 
' 'pitch of screw upon CD FxH 

FIG. 229. 




The guiding screw being right-handed, the above arrangement 
is suitable for cutting right-handed screws. 

To cut a left-handed screw it is essential that AB and CD 
shall revolve in opposite directions. 

Now AB revolves with the mandrel of the lathe, and therefore 
the direction of the rotation of CD must be reversed. This is 
effected by interposing an idle wheel between H and K, which re- 
verses the motion of the guide screw, CD, and makes the nut 
travel in the reverse direction. 

There is a double slot or groove upon the arm which carries 
K, in order to allow the adjustment of this idle wheel. 



208 



Elements of Mechanism. 



A set of change wheels is furnished with these lathes, and a 
table indicates the wheels required for cutting a screw of any given 
number of threads to the inch. The screw upon CD having two 
threads in an inch, the numbers of teeth to be assigned to E, F, 
H, K, are given in the table of which a specimen is subjoined. 



No. of threads per inch 


* 


F 


K 


H 


12 


60 


90 


2O 


120 


iH 


60 


85 


2O 


90 


13 


90 


90 


2O 


130 


34 


60 


90 


2O 


90 


is! 


80 


IOO 


2O 


no 


14 


90 


90 


2O 


140 



Ex. Let the pitch of the screw upon CD be \ an inch, and 
let it be required to cut a screw of -j^-inch pitch upon AB, or a 
screw with 13 threads to the inch. 

Here e = , which is satisfied in the following manner : 
E x K go x 20 



F x H 130x90' 

In the case of the micrometer screw with 150 threads to the 
inch, mentioned in the introductory chapter, a lathe is employed 
for cutting the thread. 

The guiding screw has 50 threads to the inch, and the mandrel 
of the lathe rotates faster than the guiding screw in the proportion 
of 3 to i. 

This change of velocity is effected by two wheels having these 
proportions, and connected by an intermediate wheel, the position 
of the centre of which can be altered so as to suit principal wheels 
of different sizes. 

The cutter which shapes the thread has a fine pointed edge, 
and the screw is nearly finished in the lathe, but is finally rendered 
perfect in form by screwing it through a pair of dies. This latter 



Wheels in Trains. 20$ 

operation has a tendency to alter the pitch of the screw by per- 
manently stretching the metal of which it is made, and should 
therefore be resorted to as little as possible. 

A screw with 150 threads to the inch, and furnished with a 
graduated head reading off to hundredths of a revolution, would 

measure a linear space of or T-^Tnrth of an inch. 

150 x 100 

ART. 173. We pass on now to an enquiry into the construe* 
tion of a train of wheels for any given purpose ; and here it is ne- 
cessary to point out that mechanicians are tied down by practical 
considerations, whereby it often happens that an arrangement, 
which is quite simple and feasible in theory, would nevertheless 
prove utterly absurd if any attempt were made to carry it out in 
practice. One very simple example will explain what we mean. 

Suppose it to be required to communicate motion from one 
axis, A, to another, C, and that C is to make 60 revolutions while 
A makes i revolution, as in the clock train. If A be made to 
drive C directly, it is clear that the number of teeth upon A must 
be 60 times as great as the number upon C, so that if C have 8 
teeth, A must have 480 teeth. 

This would involve the use of two wheels side by side, one of 
which was 60 times as large as the other, to say nothing of the 
practical difficulty of dividing the larger wheel so as to form the 
teeth, and accordingly no such combination is to be found in any 
clock train. 

But the insertion of an intermediate axis relieves us at once 
from the difficulty. 

Place such an axis, which we may call B, between A and C, 
and fasten upon it two wheels of 8 and 60 teeth respectively, give 
64 teeth to A and 8 teeth to C, and the value of the train becomes 

~ ~, or 60, the necessary result being obtained with perfect 

O X O 

ease and complete simplicity of construction. 

ART. 174. We see, then, that this problem of connecting 
two axes by a suitable intermediate train of wheels is an arithme, 
tical problem which may, of course, in some cases prove extremely 
troublesome, and may demand a considerable amount of arithme- 
tical ingenuity. 

P 



2io Elements of Mechanism. 

The value of e being assigned as a fraction, the only thing to 
be done is to resolve the numerator and denominator into their 
prime factors, and then to compose the best train which may sug- 
gest itself. 

Thus, let it be required to connect two axes so that one shall 
revolve n times while the other revolves once. 
Assume some value for n, say 720. 

Then e= n= 10x9x8, 
_ Sox 72 x64 
.-8x8x8 ' 
which gives a probable solution for the train. 

If any of the factors appear unmanageably large, we may ap- 
proximate to the value of e by continued fractions, and seek other 
factors which present less difficulty. If the value of e be an in- 
teger, we have seen that it must still be split up into factors, and 
must be further multiplied and divided by the numbers of teeth 
in each pinion. 

Thus, suppose the two axes are to be connected whereof one 
revolves in 24 hours, and the other in 365 days 5 hours 48 minutes 
48 seconds, as in Mr. Pearson's orrery. 

Since 24 hours = 86400 seconds, 
and 365 days 5 hrs. 48 min. 48 sec. = 31556928 seconds. 




IQX 9 X5 

Here 269 is an inconveniently large number, and 5 is certainly 
too small. The wheel of 269 teeth cannot be got rid of without 
altering the entire ratio, but the pinions of 9 and 5 teeth may be 
changed into others of 18 and 10 teeth. 



10 x 10 x 18 

We might have approximated to e by an algebraical process 
and have derived the fraction 

94063 3i<;i:6Q28 

: J > as representing * ?p - - very closely, 



Wheels in Trains. 2 1 1 

But 

260 4x5x13 ' 

_44X 89x97 
8x 10 x 13' 

which avoids the higher number 269, and corresponds to a period 
of 365 days 5 hours 48 min. 55-4 sec. 

Thus we have a train in which the numbers are suitable. 
Every possible arithmetical artifice Is resorted to in cases of 

this kind, and the ratio 2 ^ has been dealt with after the following 

manner, in virtue of the discovery that 269001 9x9x9x9x41, 
and it is not very difficult to get upon the necessary track, for we 
see at once that 269001 is divisible by 9 because the sum of its 

digits is so divisible, and again 9 001 = 29889, which is again 

divisible by 9 for a like reason, and thus we soon arrive at the last 
quotient after all the successive divisions by 9, viz., 41. 

Since 2 - 6 9 = _J62_, 
i 10 x 10 x 10 

269001 

= very nearly. 

10 x 10 x 10 

The numerator can now be split up into 3 factors, which will 
express the numbers of teeth in the 3 wheels of a train, and we 
may consider that 

,= 1 6 _9 = 269001 earl 

I 10X10X10 

8__x 81 X4 1 
10 x 10 x 10' 

an approximation which would introduce an error of only one 
revolution in 269000. 

ART. 175. It is also a matter of enquiry to ascertain the 
smallest number of axes which may be concerned in the trans- 
mission of any required motion, since we do not want to employ 
more wheels than are necessary. 

The smallest number of teeth which are to be allowed upon a 
pinion must be given, as well as the largest number to be allowed 
upon any wheel 



212 Elements of Mechanism. 

Suppose that no pinion is to have less than 6 teeth, and no 
wheel more than 60, and let us trace the values of e. 

With two axes e = = 10. 
6 

If the numerator be diminished, or the denominator be in- 
creased, the resulting value of e is lessened, or, in other words, 10 
is the greatest possible value of e when two axes are employed. 

With three axes the greatest value of e is - x -, or 100, and 

with four axes it is 1000, and so on. 

Let e have some value between 10 and 100 ; we observe that 
three axes will suffice, and that each wheel must have less than 60 
teeth in order to reduce e from 100 to 60. 

Thus e =^ x 4= 60. 
6 6 

Again, let e = 3-A-ss 12 if, and suppose 180 and 12 to be the 

O 

limiting numbers of teeth upon a wheel and pinion respectively. 
In the train which is about to be composed, we shall now find 
that this extension of the limits of the numbers of teeth upon the 
respective wheels and pinions will give us the power of arranging 
the train without increasing the number of axes. 

Here =1, 



and x =15x15 = 225. 

12 12 

Now 1 2 if is less than 225, and therefore three axes will suffice, 
as in the train represented by 

c __ i8c? x 146 
18 12 ' 

We may work out this arithmetical reasoning by the use of 
symbols, and then our solution will apply to every case which can 
occur. 

Assume now that p represents the least number of teeth upon 
a pinion, w the greatest number upon a wheel, and let x represent 
the number of fractions in e. 

If all the fractions making up the value of e were equal to each 



Wheels in Trains. 213 

other and had the greatest admissible value, then e would reach its 
limiting value, and we should have 

e= x H x . . . . to * factors = (^ 

P P P \p 

whence log e x (log w log /), 

log e 

""log w - log/ 

Now x will probably be a fraction, in which case the next in- 
teger greater than x + i will represent the required number of 
axes. 

Ex. Let e = $-$-, w = 180, p = 12, 

. ^_ . x = log 365 - log 3 
"/ log 15 

= i + a fraction. 

Now the integer next greater than x -f i is 3, therefore 3 axes 
will be required. 

We observe that it is not necessary to find the actual value of 
x, but simply to ascertain the integer next greater than it. 

ART. 176. It is sometimes a matter of enquiry how often any 
two given teeth will come into contact as the wheels run upon 
each other. We will take the case of a wheel of A teeth driving 

one of B teeth where A is greater than B, and let ~ when re- 

B b 

duced to its Lowest terms. 

It is evident that the same points of the two pitch circles 
would be in contact after a revolutions of B or b revolutions of A. 

Hence the smaller the numbers which express the velocity ratio 
of the two axes, the more frequently will the contact of the same 
pair of teeth recur. 

i. Let it be required to bring the same teeth into contact as 
often as possible. 

Since this contact occurs after b revolutions of A or a revolu- 
tions of B, we shall effect our object by making a and b as small 
as possible, that is, by providing that A and B shall have a large 
common measure. 

Ex. Assume that the comparative velocity of the two axes is 



2 1 4 Elements of Mechanism. 

intended to be nearly as 5 to 2. And first make A = 80, B = 32, 
in which case we shall have 

4=^=5 exactly, 
B 32 2 

or the same pair of teeth will be in contact after five revolutions of 
B, or two revolutions of A. 

2. Let it be required to bring the same teeth into contact as 
seldom as possible. 

Now change A to 81, and we shall still have -= very nearly, 

B 2 

or the angular velocity of A relatively to B will be scarcely dis- 
tinguishable from what it was originally. But the alteration will 

effect what we require, for now = , which is a fraction in its 
B 32 

lowest terms. There will therefore be a contact of the same pair of 
teeth only after 81 revolutions of B or 32 revolutions of A. 

The insertion of a tooth in this manner was an old contrivance 
of millwrights to prevent the same pair of teeth from meeting too 
often, and was supposed to ensure greater regularity in the wear of 
the wheels. The tooth inserted was called a hunting cog, because a 
pair of teeth, after being once in contact, would gradually separate 
and then approach by one tooth in each revolution, and thus ap- 
pear to hunt each other as they went round 

The clockmakers, on the contrary, appear to have adopted the 
opposite principle. 

Finally, we would remind the reader that everything which we 
have said here about wheels in trains is true, whatever be the direc- 
tions of their axes. We only care to know the relative sizes of the 
pitch circles and the directions in which they turn : any part of 
the train may be composed of bevel wheels without affecting our 
results. 



Aggregate Motion. 2 r $ 




CHAPTER VII. I / 

IZZT" 

AGGREGATE MOTION. ' / / 

ART. 177. We have seen that every case of the curvilinear 
motion of a point is of a compound character, resulting from the 
superposition of two or more rectilinear motions. 

It often happens in machinery that some revolving wheel or 
moving piece becomes the recipient of more than one independent 
motion, and that such different movements are concentrated upon 
it at the same instant of time. 

The motion is then of a compound or aggregate character, and 
we propose to classify under the head of ' Aggregate Motion ' a 
large variety of useful contrivances. 

We commence with two or three simple examples. 

The well-known frame called Lazy Tongs is a contrivance 
depending upon aggregate motion. 

The rapid advance of 
the ends A and B is due to 
the fact that these points 
are the recipients of the 
sum of the resolved parts 
of the circular motion which 
takes place at each angle. 

Consider the angular joints at the ends of the first pair of bars 
which carry the handles : these ends of the bars describe circles, 
just as the points of a pair of scissors would do. Either of these 
motions in a circular arc may be resolved as in Art. 7 and one 
of the components so obtained will be carried to the end of the 
combination. The same thing happens at every joint of the series, 
and thus A and B receive the aggregate of all these separate 
movements. 

A wheel rolling upon a plane is a case of aggregate motion ; 




2 1 6 Elements of Mechanism. 

the centre of the wheel moves parallel to the plane, the wheel 
itself revolves about its centre, and these two simple motions give 
the aggregate result of rolling. 

Thus, in the case of the driving-wheel of a locomotive, each 
point on the tyre becomes a fulcrum upon which the rest of the 
wheel turns, and is for an instant absolutely at rest. The centre of 
the wheel has the velocity of the train, while a point in the upper 
edge moves onward with twice that linear velocity. Simple as 
this matter is, it puzzles some persons when they first think 
about it. 

In the same way, if a beam of timber be moved longitudinally 
upon friction rollers, the travel of the beam will be twice as great 
as that of the rollers. 

So, again, in moving heavy guns, the men employ what is 
called a wheel purchase ; that is, they fasten one end of a rope to 
the spoke of a wheel of the gun-carriage, and make the rope run 
round the rim. This gives them the leverage of the spokes of the 
wheel, and the power exerted is exactly one-half of what it would 
be if the rope were attached directly to the axis of the wheel, in 
virtue of this principle that the linear velocity of the upper part 
of the rim is twice that of the centre of the wheel. 

In some printing machines the table is driven by a crank and 
connecting rod, and the length of its path may be doubled by 
applying the principle under discussion. 

Here a wheel, Q, is attached to the end of the connecting rod 
PQj so that it can turn freely on its centre, Q. 

FJG 23I . Let the wheel revolve 

between the two racks A 
and B, whereof A is fixed 
to the framework of the 
machine, while B carries 
the reciprocating table. 

The rack B receives 
the motion of Q in its 
twofold character, and 
moves through exactly twice the space that it would describe if 
connected simply with the point Q. 

The size of the wheel makes no difference in the result, for in 




Differential Pulley. 



217 



all cases the velocity of a point in the upper edge will be twice 
that of the centre. 

ART. 178. We may confirm our views of the nature of rolling 
motion by seeing what would happen if the fulcrum, round which 
the wheel turns, were raised above the level of the road. 

We have now a contrivance by which a carriage may be made 
to move faster than the horse which draws it, a startling method 
of stating the fact which has been sometimes adopted. The in- 
ventor was a Mr. Saxton, who patented a Differential Pulley in the 
year 1832 (No. 6,351), with a view of obtaining great speed in 
railway caniages propelled by a rope. By the use of this inven- 
tion, the consumption of the rope, proposed to be wound up at a 
stationary engine house, would be much less than if the carriage 
were attached in the ordinary way. 

Let two wheels of different diameters (say as 6 to 7) be centred 
on a common axis at C, and be fastened together, and let an 

FIG. 232. 




endless rope be wound round the wheels and pass over pulleys at 
E and F in the manner shown in the diagram, the rope taking a 
turn round each of the pulleys. 

Conceive now a pull to be exerted on the rope at A, in the 
direction AF, then the tension of the string will cause an equal 
and opposite pull to be felt at B in the direction BE, and thus the 
compound pulley has a tendency to turn about D, the middle 
point of AB. 

This tendency in the pulley to turn about the point D causes 
the linear motion of C to be very much greater than that of any 
point in the rope : for example, when B moves through a small 



2 1 8 Elements of Mechanism. 

space B^, the centre C will advance through G", which upon our 
supposition is thirteen times as great, so that when one yard of 
rope is wound up, the carriage will have travelled through 13 
yards. 

The carriage may be at once stopped by disconnecting the 
pulleys. 

ART. 179. The differential screw is another instance of aggre- 
gate motion, and is a favourite with writers on mechanics, inasmuch 
as it gives theoretically a mode of obtaining an enormous pressure 
by the action of a comparatively small force. 

It is constructed on the following principle : two screw threads 
of different degrees of inclination are formed upon the same 
spindle AB, the spindle itself passing through two nuts, whereof 
one, E, is part of a solid frame, and the other, D, can slide in a 
groove along the frame. Let P, Q represent the pitches of the 
screws at E and D ; then upon turning AB once the nut D is 
carried forward through a space P, and is brought back again 
through a space Q : it therefore advances through the difference 
of these intervals. (Fig. 233.) 




There is a form of the differential screw described in the 
fifteenth volume of the 'Philosophical Transactions,' which is 
known as Hunter's Screw. Here one screw is a hollow tube acting 
as a nut for the second screw in the manner shown in fig. 234. The 
smaller screw is attached to a piece D sliding in the frame, and is 
not allowed to rotate : upon turning the screwed pipe AB, the 
piece D will move through a space equal to the difference of the 
pitches of the two screw threads. 

If one screw thread were right-handed and the other left- 
handed, the nut would travel through a space, P + Q, upon each 
revolution. 

ART. 1 80. A right and left-handed screw are often seen in 
combination, for the purpose of bringing two pieces together. 



Aggregate Motion. 



219 



FIG. 235. 




There is a very common instance in the coupling which is used to 
connect two railway carriages. Upon swinging round the arm AB, 
the screws which are moved by it bring the nuts E and F at the 
ends of the coupling 
links closer together, or 
cause them to separate. 
This is obviously a most 
convenient arrangement. 

The lever arm and 
weight at B serve a two- 
fold purpose: they enable 
the railway servant to 
screw up the combina- 
tion easily, so as to put a pressure upon the buffer-springs, and 
the weight B prevents the screws from shaking loose during the 
running and vibration of the train. 

There is another instance of the use of a right and left-handed 
screw in combination which is found in the valve-motion of 
Nasmyth's steam-hammer. 

Here a right and left-handed screw are placed side by side, 
and are connected by spur-wheels so that they rotate in opposite 
directions. Two nuts fastened together engage with the separate 
screws, and both rise and fall at the same time, being both ad- 
vanced in the same direction by screws which rotate in opposite 
directions. 

ART. 1 8 1. Any system of pulleys will form an example of 
aggregate motion. 

Taking the single movable pulley in fig. 236, 
it is apparent that when W is raised one inch, the 
centre of the block rises an inch, and therefore 
the end P of the line DP is shifted one inch. 

But at the same time the circular sheave of the 
pulley runs upon the line AB, just as a wheel runs 
upon a plane, and by turning on its centre until an 
additional inch of string has come in contact with 
it, will transfer the end P through another space of 
one inch, whereby, on the whole, P moves through 
two inches. 



FIG. 236. 




220 



Elements of Mechanism. 



ART. 182. Another contrivance for lifting heavy weights by a 
small expenditure of power is the Chinese Windlass. 

Here a rope is coiled in opposite directions round two axles A 
and B, of unequal size : the rope is consequently unwound from 
one axle while it is being wound up 
Flt " 2 ^ v 7 ' by the other, and the weight may 

rise as slowly as we please. 

Let R, r be the radii of the axles, 
then W moves through TT (R r) 
upon each revolution of the axles. 

The practical objection to this 
windlass consists in the great length 
of rope required during the opera- 
tion. 

In the ordinary windlass the 
amount of rope coiled upon the 
barrel represents the height through 
which the weight is raised, whereas 
here we begin by winding as many 
coils on the smaller barrel as the 
number of turns which we intend to 
make with the winch -handle, and then at the close of every turn 
a length of rope equal to 2?rR is coiled upon the larger barrel, by 
which expenditure the weight has only been lifted through v (R r). 
Ex. Let R= u, r= 10, then the amount of rope wound up 
in any number of turns bears the same proportion to the space 
through which the weight is raised that 22 bears to i. 

This is a sufficient commentary on the invention regarded as a 
practical contrivance. 

ART. 183. The object of Westorfs Differential Pulley -block 
is to avoid this difficulty about the expenditure of rope. In the 
Chinese Windlass, one end of the rope is supposed to be fastened 
to the axle A, and the other end to the axle B. If, however, these 
two ends were brought together, the supply of rope necessary for 
B might be drawn from that coiled upon A, and the expenditure 
would be really 2* (R r). There would be many inconveniencies 
attending this arrangement in practice, but it has been put into a 
working shape in the manner shown in the drawing. 




Differential Pulley. 



221 



In Weston's pulley-block there are two pulleys A and B, nearly 
equal in size, turning together as one pulley, and forming the 
upper block : an endless chain supplies the place of the rope, and 
must of course be prevented from slipping by projections which 
catch the links of the chain. The power is exerted upon that 
portion of the chain which leaves the larger pulley, the slack hangs 
in the manner shown in the sketch, and the chain continues to 
run round till the weight is raised. The combination is therefore 
highly effective. 





Of the detached sketches, one shows a section of the differen- 
tial block, and the other is intended to explain the mechanical 
principle involved. 

Let C be the centre of the compound block, draw the hori- 
zontal diameter ABCDE, and let it be noted that the string or 
chain is unable to slip upon the surface of either pulley. 

Let P represent the tension of AP, 
W the weight raised, T the tension of the string or chain at E, 




222 Elements of Mechanism. 

Also let R, r, be the radii of the respective blocks, 
Then TxCE=Px AC + T x BC, 

or TxR=PxR + Tx?" . . . (i) 
Also 2 T=W ..... .... (2) 

/. W(R-r)= 2 PxR, 



Ex. 



2x15 

or W= 3 oP. 

Regarding the question as an application of the principle of 
work, the diagram sets out the calculation as follows : 

Since the diameter of the pulley A is 15, we shall assume that 
a point in the chain passing over A moves through a space 30, or 
that P has a motion 30. Also the diameter of B is 14, whence it 
follows that the chain passing over B will have a motion 28 in the 
opposite direction. Note. In fig. 238, taken from the 'Anderson ' 
series, the symbol ' Mo.' stands for the word ' motion.' 

Hence the motion of the chain round the pulley which sup 

ports W is (30-28), and the motion of W itself is 3 2 = i. 

Hence motion of W : motion of P : : i : 30, 
or W : P::3o : i. 

ART. 184. The subject of Epicydic trains will now occupy 
our attention, and we shall discuss some of the most useful 
applications of that peculiar arrangement of wheelwcrk which is 
technically so designated. 

An epicyclic train differs from an ordinary train in this par- 
ticular : the axes of the wheels are not fixed in space, but are 
attached to a rotating frame or bar, in such a manner that the 
wheels can derive motion from the rotation of the bar. 

There are certain fundamental forms which consist of trains of 
two or three wheels ; the first wheel of the train is usually con- 
centric with the revolving arm, and the last wheel may be so 
likewise. 

It should, however, be understood that any number of inter- 



Epicyclic Trains. 



223 



mediate wheels may exist between the first and last wheels of the 
train, and that the wheels in the train may derive the whole of 
their motion from the arm ; or they may receive one portion from 
the arm and the remainder from an independent source. 

The elementary form of a train is exhibited in the annexed 
diagram, and the peculiarities which result from compounding any 
independent motion with that which arises from the rotation of 
the arm will demand some careful and attentive study. 

Here it will be seen that the wheel B, or the wheels B and C, 
are attached to a bar which is capable of revolving about the centre 

FIG. 239- 




of the wheel A, the axis of this latter wheel being firmly held in 
one position. 

ART. 185. In order to understand movements of this kind 
let us take a simple case to begin with. 

Suppose that there were only two wheels in the train, viz., A 
and B, and let A be locked so that it cannot rotate ; suppose, 
further, that A has 45 teeth, and that B has 30 teeth, and let us 
inquire how many rotations B will make while the arm is carried 
round once. 

We might at first imagine that the wheel B would rotate %% or 
f times by running round upon A ; but this is only a part of its 
movement. The wheel B has also been carried round in a circle 
about A by reason of its connection with the arm, and having 
turned upon its axis once more on that account, it has really made 
% turns, instead of |, during one revolution of the arm. 

In confirmation of this view, let us consider the case of three 
wheels, A, B, and C, whereof A and C are equal. As the arm 
goes round, we conclude that C will turn once in the opposite 
direction to the arm by the rolling of the wheels, and it will turn 
once in the same direction as the arm by reason of its connection 



224 



Elements of Mechanism. 



therewith ; the aggregate result being that C will be carried round 
in a circle without rotating at all upon its own axis. 

FIG. 240. 




The motions of the wheels B and C in an epicyclic train are 
shown in the sketch. The arm is supposed to have revolved 
through an angle of 45, and it will be seen that B has turned 
round through a right angle, while C has not rotated at all. 

We propose now to examine the motion by the aid of analysis. 

Remembering that there may be any number of wheels in the 
train, of which A is the first, and L the last wheel. 

Conceive that the arm makes a revolutions! dufi the ^ 

the first wheel A makes ,// revolutions > iod Qf d 

the last wheel L makes n revolutions f 
and let e be the value of the train. 

Then the first wheel makes (ma) revolutions relatively to the 
arm, and the last wheel makes (n d) revolutions relatively to the 
same arm, or, in other words, L makes (na) revolutions for 
(m a) revolutions of A. 

Recurring to our definition of the value of a train (see Art. 163), 
we at once deduce the equality 

n a 

e = - 
ma 

There are three principal cases to consider ; 
i. Let A be fixed, or ;;/ = o, 



or n = a (1 ~e) and a = -- 



Epicyclic Trains. 225 

Let L be fixed, or =o, 



otm a(i--\ and = 
V e) ei 



3. Let neither A nor L be fixed, 

we have now the formula 

m a 

whence emea=.n a, 
or n=me+(ie)a. 

In applying these formulae we must remember that e is positive 
when the train consists of 3, 5, or an odd number of wheels, and 
negative when there are 2, 4, or an even number of wheels. 

Ex. i. Let there be two equal wheels, A and B, in the train, 
and conceive A to be locked, or let A be a dead wheel^ as it is 
termed. 

Here m = o, and e = i, 

or the wheel B makes two rotations for each revolution of the 
arm. 

Ex. 2. Let there be three wheels, A, B, and C, whereof A and 
C are equal, and let A be a dead wheel as before. 

Here ;// = o, e = i, 
whence n = a (i <?) =a(i i) = o, 
or C does not turn on its axis at all. 

Ex. 3. Take the case first considered, where A is a dead wheel 
and has 45 teeth, and where B has 30 teeth. 

Here 



the result arrived at by general reasoning. 

ART. 1 86. The Sun and Planet Wheels were invented by 
Watt, and were used to convert the reciprocating motion of the 
working beam of an engine into the circular motion of the fly- 
Q 



220 



Elements of Mechanism. 



wheel. We have already referred to this invention in Aft, 37, 
and have explained the object which it was intended to fulfil. 



FIG. 241. 




The drawing shows Watt's invention as specified in a patent of 
1781 (No. 1,306). CB is the working beam, and AB is the spear 
or connecting rod ; E is a wheel fixed upon the end of the shaft 
or axis F, which receives the rotatory motion which is communi- 
cated to it by a second wheel, firmly fixed to AB in such a man- 
ner that it cannot rotate. Behind EB there is a heavy wheel, GG, 
having a groove or circular channel around its circumference, into 
which a pin at the back of A enters. The wheels A and E are 
thus kept in gear, and some such precaution is indispensable, but 
instead of the wheel with the groove and pin there may be a link 
connecting A and F. The construction having been described, 
the specification states that in the working of the engine the con- 
necting rod pulls the wheel A up and down ; and since its teeth 



Sun and Planet Wheels. 227 

are locked in the wheel E, and it cannot turn upon its own axis, 
it cannot rise or fall without causing E to turn upon the axis F. 
When the two wheels A and E have equal numbers of teeth the 
wheel E makes two revolutions on its axis for each stroke of the 
engine. 

We may explain the peculiarity as follows. If the discs E and 
A were fastened together at the point a, and E were to FIG 
make half a revolution, A would come into the position 
A', and the direction of the arrow marked upon it would 
be reversed. But in the actual motion this arrow retains 
its first direction, and in order to recover it, the disc A' 
must again rotate through 180, and must carry E round 
through another half-revolution : so therefore when we 
recur to the arrangement invented by Watt, E will make 
a complete revolution while A descends from the highest 
to the lowest position, or travels half-way round it. 

If we were to apply our formula ( e = ?-J? ] we should 

V m aj 

make L the dead wheel, in which case n = o, and e = i, 

~-^ a 

ma 

.'. ma=a, or m=2a, 
which is the result already arrived at. 

ART. 187. Ferguson's Paradox is obtained by placing three 

wheels upon the axis which usually carries C, and making these 

wheels very nearly equal to each other, and very nearly equal to A. 

Thus let A have 60 teeth, and let the numbers of teeth upon 

E, F, G, be 6 1, 60, 59, respectively. 

FIG. 243. 




The number of teeth upon B is immaterial, and the wheel A 
is fixed to the stud upon which it rests, and does not rotate with 
the arm, so that m = o throughout the motion. 
Q 2 



228 Elements of Mechanism. 

1. Taking the general formula e ^- , we have, in the case 

m a 

of the wheel E, 

tn = o, and e = , which is less than unity. 
61 

n a 66 

a 61 
.'. 6 1 n 6 1 a 60 a, 

whence n = , and is positive. 
61 

2. For the wheel Fj e=~ = i, 

60 

.'. n a =a, or n = o. 

3. For the wheel G ; e = , which is greater than unity, 

59 



whence # = , and is negative. 

So that when the arm is made to revolve round the locked or 
dead wheel A, the wheel E turns slowly in the same direction 
as the arm, F remains at rest, and G moves slowly in the reverse 
direction. This combination has formed a rather popular me- 
chanical puzzle. 

The general result deducible from Ferguson's paradox is the 
following : 

Let there be a train of three wheels, viz., A, B, and C, and let 
A be a dead wheel, and greater than C, then the rotation of the 
arm in a given direction will cause C to rotate in the opposite 
direction in space. 

Whereas when A is a dead wheel and less than C, the rotation 
of C will take place in the same direction as that of the arm. 

ART. 1 88. Problem. Let it be required to draw an exact 
straight line by means of an epicyclic train. 

Conceive that ACQ represents a bar made up of two equal 
straight lines AC, CQ, the point A being a centre of motion, and 
the point C representing an ordinary rule joint. 



Straight Line Motion. 



229 



Let ACQ be placed so as to coincide with the straight line Ay. 
Then it has been proved that if AC turns through an angle in one 
direction, the while CQ turns through the same angle in the oppo- 
site direction, the point Q will trace out a portion of the straight 
line. 

The student should refer to Art. 117, where it is shown that 
if Ax be drawn at right angles to Ay, and QC be produced to 
meet Ax in R, the triangle QAR lies always in a semicircle whose 
centre is C, and the property above stated is a necessary conse- 
quence of the, sliding of QR between Ax and Ay. 

FIG. 244. 





It only remains for us to demonstrate that the required motion 
of CQ may be obtained by mounting that line upon the last wheel 
of an epicyclic train with three axes. 

Conceive that three wheels, viz., A, B, and C, are mounted on 
the arm AC, and are mutually in gear. 

Let it be arranged that the axis of the first wheel shall coin- 
cide with A, and that of the last wheel with C, and let the diameter 
of C be half that of A. The size of B is immaterial. 

We have now to apply the formula e = ^- a - 

m-a 

Here there are three wheels, and the number of teeth on C is 
half that on A, whence e = 2. 

Also let A be a dead wheel, therefore m = o. 

n a 

.'. 2 = - - , or n = a, 
o a 

whence C rotates in a backward direction at the same rate as the 
arm rotates in a forward direction. 



230 



Elements of Mechanism. 



It follows that the point Q will describe the straight line Ay. 

In like manner if QC be produced to R, such that CR=CQ, 
it is apparent that the point R will describe the straight line A.*, 
which is perpendicular to Ay. Thus the extremities of a straight 
line QR, which is carried by the wheel C, and bisected at the 
centre C, will describe straight lines intersecting at a right angle. 

As soon as the nature of the primary motion is understood. 
the proposition now considered becomes a simple deduction 
therefrom. 

ART. 189. It will of course be understood that models may 
be readily constructed for illustrating the fundamental propositions 
in this branch of mechanism without employing toothed wheels. 
As an example we refer to the diagram, which is taken from a 
model and is intended to exhibit the results obtained by a train of 
two or three spur wheels. 

FIG. 245. 





It will be seen that there is a fixed upright pillar carrying an 
arm AC, centred at A, and having upon it two round discs or 
pulleys, A and C. These pulleys are equal, but they may be un- 
equal and of any convenient dimensions, and they are further 
connected either by crossed or open bands. 

When the band is crossed we have an equivalent for two spur 
wheels in gear, and when it is open the combination is the same 
as that of three spur wheels. 



Epicydic Trains. 231 

Taking the case where A is a dead wheel and equal to C, the 
result of moving the arm through a given angle is shown in the 
sketch. The wheel C .carries a pointer D, and the faint dotted 
lines show the starting position of the arm and of the pointer. 
When the strap is open, CD remains vertical throughout the mo- 
tion, but when the strap is crossed it rotates in the same direction 
as the arm with a velocity ratio of 2 to i. It follows that the 
angle DC/ is equal to twice the angle CAC'. CL-*-^ 

ART. 190. Numerous models, designed for illustrating simple 
astronomical problems, may be formed by properly arranged discs 
and bands. 

Thus, it is quite easy to exhibit mechanically the phases of the 
moon. 

For this purpose a small silvered ball M, representing the 
moon, is attached by a pipe or hollow stem to a bar capable of 
being carried in a horizontal plane round a fixed ball, E, intended 
to represent the earth. Underneath the arm are the driving 

FIG. 246. 




pulleys, which consist of '(i) a dead wheel A, and (2) a pulley C, 
equal to A, and connected with it by an open band. A hemi- 
spherical black cap, which obscures exactly one half of the surface 
of M, is attached to a wire passing through both M and the hollow 
stem, and fastened to C, so as to form part thereof, and to move 
with it. The sun is supposed to lie on the left-hand side of the 
diagram, and as the moon is carried round E, the shaded portion 
lies always to the right hand. It is apparent that the motion of 
the cap, or of C which directs it, should be that of the third wheel 



Q. 



232 Elements of Mechanism . 

F in Ferguson's paradox, for which purpose we require that A 

should be equal to C, and that 

' G y^ 7 ' the band connecting them should 

,.'-"' "NS,, be open, the wheel A being a 

& (ft dead wheel. 

As the arm revolves, the disc 
\ C moves round in a circular path 
^ without at all rotating upon its 

p own axis, and the hemispherical 

J / cap takes the various positions 

shown in the sketch, imitating 
jjl thereby the shadow which would 

"--.. (*---"'''' ke caused by a luminous body at 

a great distance to the left of the 
globe E. 

ART. 191. An epicyclic train may also be formed by the use 
of three bevel wheels, A, B, arid C, connected as in the figure, 
and we find now the peculiarity that the 
wheels A and C turn in opposite direc- 
tions. 

The formula already investigated ap- 
plies equally in this case, and some of 
the results to be obtained are extremely 
useful. 

Our first example shall be an arrange- 
ment whereby the continuation of a piece of shafting may be 
made to rotate twice as fast as the first portion of it. This forms 
a simple and easy method of obtaining an increased velocity in a 
revolving piece, and is used to rotate the coils in some magneto- 
electric machines. 

Thus, let A be a dead wheel, and let B ride loose upon an 
arm which itself is rigidly attached to the first portion of the 
shaft, namely, that passing through A, the wheel C being keyed to 
the other portion which is required to revolve with increased 
rapidity. As the arm carrying B goes round A, we can easily see 
that if B were not allowed to rotate at all it would still carry C 
round once, and that its rotation upon the dead wheel A carries C 
round a second time, and thus we have an exact reproduction of 




Differential Motion. 



233 



FIG. 249. 



the motion of the two equal spur wheels, one of which is a dead 
wheel, and obtain two rotations of C for each revolution of the 
arm carrying the intermediate wheel B. 

We may of course apply the general formula in the case of 
bevel wheels just as in that of spur wheels, and the expression 

e = gives the result obtained before. 
m a 

Thus m = o, e = i, since A and C revolve in opposite di- 
rections. 

n a 

.'. i = , or n=2a, 

a 

whence C goes round twice for each revolution of the arm. 

ART. (fg^ An illustration may now be taken from the cotton 
mills of Dmcashire. 

During the process of the manufacture of cotton yarn or 
thread, it is essential to wind the partially twisted fibre upon bob- 
bins, and at the same time this 
fibre, or roving, must not be sub- 
jected to any undue strain. 

The fibre is delivered to the 
bobbins at a uniform rate, whereas 
the bobbins get larger as they fill 
with the material, and hence the 
winding machinery must be so 
contrived that the rate of revolu- 
tion of the bobbin shall slowly 
decrease upon the completion of 
each layer of the fibre. 

In the year 1826 Mr. H. 
Houldsworth patented an inven- 
tion which solves the problem 
of the bobbin motion in the most 
complete and satisfactory manner. 

In the preceding article we 
have supposed the wheel B to be 

carried by an arm which is capable of revolving round the axis 
AC. The better way, however, of suspending B for our purpose is 
to attach it to the face of a spur wheel, H, as in fig. 249. 




234 Elements of Mechanism. 

Let this be done, and let A be connected with the driving 
shaft of the engine, so that its rotation shall necessarily be constant. 

Tf now some independent motion be imparted to the wheel H, 
the result may be calculated from the formula. 

Here A, B, C are equal in size, and C rotates in a direction 
opposite to that of A, 



which gives the analytical relation between the angular velocities 
of A, C, and H. 

If we examine this formula, we shall comprehend that the 
velocity of C may be reduced by altering the velocity of H. 

1. For let a= m, or let A and H turn at the same rate, 

then n + m = 2 a = 2M, 
.". n = m, or C has exactly the same motion as A. 

2. Let a = 3, that is, let H make three revolutions while A 

4 
makes four, 



3. Let a = , in which case H makes one revolution for two 
revolutions of A, 



.'. n = o, or C stops altogether. 

We have taken extreme cases, from which it appears that when 
the velocity of the arm is made less than that of A, the velocity 
of C is reduced in a twofold degree. 

Generally, let a = m 

2m 2m 

.'. n = 2a m = 2m m = m > 

x x 

or the rate of diminution of n is twice that of a. 

It now becomes easy to obtain any required reduction in the 



Differential Motion. 



235 



velocity of C. A reduction in the velocity of H must first be 
effected by shifting a driving strap along a conical pulley, and 
the velocity of C will be reduced twice as much as that of H. 

Mr. Houldsworth's invention consists, therefore, in imparting 
to the wheel C two independent motions which travel by different 
routes, and which, after combination in the manner just investi- 
gated, are capable of producing the desired differential motion. 

ART. 193. In order to fix our ideas, let us calculate the 
motion in the following ex- FIG. 250. 

ample : 

Suppose A, B, C to repre- 
sent three equal wheels, and 
let A be fixed to a shaft AD, 
which carries a conical pulley 
provided with grooves at a, b, 
c, d, e, where the diameters 
are 4, 5, 6, 7, 8. 

EF is another shaft carry- 
ing a second conical pulley 
which is the counterpart of 
the first, and terminating in a wheel F, whose diameter is half 
that of H. 

A crossed band connects the two cones, and the axis AD is 
made to revolve with a uniform velocity. 

It is required to ascertain the motion of C when the strap is 
shifted along the conical pulley. 

i. Let the strap be placed at a, the angular velocity of H will 

be that of AD, and we have a = > 




4 2 

or C moves in the opposite direction to A, and with half its 

velocity. 

e r er 

2. Let the strap be at b, the velocity of H will be ~ that of , e 

AD (here e = - x - = ~- according to Art. 163), 

T r tyH i C/// 2/7/ 

therefore a i , and n=^ m = > 
r 4 7 7 



236 Elements of Mechanism. 

hence C still moves in the opposite direction to A, but less rapidly, 
in the ratio of 2 to 7. 

3. Place the strap at c, when e increases to I, and a becomes 

equal to - ; , /. n = 2 x m m = o, 

or C stops altogether, its motion being entirely destroyed. 

4. Place the strap at d, and we have a = x - = ~~, 

im 2m 

whence n-=2a m = m = > 

that is, C and A move in the same direction with velocities in the 
ratio of 2 to 5. 

5. Finally adjust the strap at e, and the velocity of H will be 
the same as that of AD. 

Here a = m, and n = 2m m = w, 
or the motion of C is precisely the same as that of A. 

The principle of this invention may now be understood, 
although it is difficult to appreciate such a movement thoroughly 
without the assistance of a model. 

It only remains to present to the student a representation of 
so much of an actual machine as will embody the cones and the 
differential train of wheels. The diagram exhibits the manner in 
which simple elementary movements may be combined together 
so as to form a train of mechanism, the arrangement of which, 
before it is properly understood, might appear to be very complex 
and intricate (see fig. 251.) 

The operation of spinning, so far as it is carried on by the 
mechanism before us, is effected by passing a partially twisted fibre 
or roving through a tube, called a flier, attached to the end of a 
spindle, and then causing both the flier and the bobbin to rotate 
with a high velocity. Before the fibre reaches the fliers it is elon- 
gated or drawn out by a combination of rollers, moving at different 
speeds and called drawing rollers ; it is therefore of necessity fed 
on at a fixed uniform rate. 

The flier and the bobbin both rotate together, and thus twist 
the roving, but they also rotate at somewhat different speeds, by 
which arrangement it is provided that the joint operations of twist- 



Differential Motion. 237 

mg the thread and of winding it up upon the bobbin shall go on 
together. 

A bobbin with its spindle and flier is shown in the sketch. It 
will be seen that the roving passes down through the hollow ver- 
tical arm and is carried to the bobbin by a finger ; the finger is 
pressed against the bobbin by the centrifugal action of a small 
elongated piece which runs down the side of the arm, and which, 
by its tendency to get as far as possible from the axis of the spindle 
during its rotation, keeps the finger pressed against the surface of 
that portion of roving which is already wound upon the bobbin. 
This part of the apparatus has formed the subject-matter of a most 
lucrative invention. 

As the winding goes on the bobbin rises and falls, and the flier 
winds the fibre in uniform layers upon the bobbin. 

Thus the spindle and flier rotate together, and they are driven 
by skew-bevels, whereof one is shown at the bottom of the drawing. 

The bobbin rotates independently of the spindle, and is also 
driven by skew-bevels, whereof one is shown just underneath the 
bobbin. Note. As to skew-bevels, see Art. 239. 

These bevel wheels are in direct communication with the spur 
wheels marked ' to spindles ' and ' to bobbins ' in the drawing. 

It will be understood that the winding on will take place when 
the spindles and the bobbins move at different velocities, and that 
either may go faster than the other. We shall take the case in 
which the bobbins precede the fliers. 

Since the spindles with their fliers move at' a fixed velocity, 
while the bobbins are continually filling with the rovings and 
becoming larger, we infer that the bobbins will require a smaller 
amount of rotation relatively to the fliers, in order that the wind- 
ing up of the fibre, which is being fed on at a fixed rate by the 
drawing rollers, may take place uniformly. Hence, if the bobbin 
runs in advance of the flier, the speed of revolution has to be 
diminished as its diameter becomes larger. 

Refer now to the sketch, and it will be seen that the power 
may pass through the combination of bevel wheels to the three 
spur wheels placed in a line at the extremity of the 'driving 
axis' and connected with the cone marked as the 'driver.' The 
driving power then crosses over to the follower, and enters the 



238 Elements of Mechanism. 

combination of bevel wheels by the small pinion upon the axis of 
the lower cone which gears with the large spur wheel marked H, 
which latter wheel rides loose upon the driving axis. 

The combination of four bevel wheels is exactly analogous to 
that discussed in Art. 191, the two wheels B and B are equivalent 
to a single wheel, and prevent the one-sided, unbalanced action 
which would otherwise occur. 

The wheel A is fixed to the driving shaft, the wheel C rides 
loose upon it, but is fastened immovably to the spur wheel marked 
' to bobbins,' the function of which has been already explained. 

We have to prove that the combination of the two cones with 
the spur and bevel wheels is capable of gradually reducing the 
velocity of the bobbins as they fill up with the roving. 

Assume that the cones are equal in section where the strap is 
placed, then the speed of the first cone will be reduced to ^ by the 
combination of three spur wheels starting from the driving axis, 
and thus the pinion which drives H will move at | the speed of 
the driving axis. 

But H is five times as large as that pinion, hence the velocity 
of H is ^th that of the driving axis. 

The wheel H also rotates in the opposite direction to the driving 
axis. 

Take now the formula n = m e + (i e) a. 

Here e - i, since A, B, and C are equal, 

and a = as we have just shown ; 

10 



5 
6m 

Hence the speed of the bobbin pinion is to that of the flier 
pinion as 6 to 5, or 18 to 15, the negative sign merely showing 
that the loose wheel C revolves in the opposite direction to the 
driving axis. 

The student may be surprised to find that all this apparently 
reducing arrangement has ended in making the last spur wheel in 



Differential Motion. 



239 




240 Elements of Mechanism. 

the train turn faster than the driving axis ; but an explanation is 
found in the fact that the rotation of H takes place in the oppo- 
site direction to the driver, that is, in the same direction as the 
loose wheel C, and accordingly we shall find that if the velocity of 
H be reduced we shall also reduce the inequality between the 
velocity of the bobbins and spindles. 

Conceive now that the strap is shifted towards the right hand 
until the sections of the cones are in the proportion of 2 to 3; 
that is, nearly as far as the spur wheels. 

The velocity of H will be reduced two-thirds, and will become 
equal to y^th that of the driving axis. 



17 m 



IS 

2 -^ 
15 



or trie relative speed of the bobbin pinion to that of the flier 
pinion is reduced from 18 to 15, and now stands at 17 to 15. 

It is hoped that the complete action of the apparatus is now 
sufficiently explained, and there is only one refinement in con- 
struction which remains to be pointed out. It will be seen that 
the upper cone is slightly concave and the lower one convex: this 
configuration is adopted because the absolute increase in the 
diameter of a bobbin bears a ratio to the actual diameter which 
is not constant, but is continually diminishing in a small degree. 
The mechanic must not forget or overlook any material point in 
working out his design. 

ART. l(^> Epicyclic trains may be employed to produce a 
very slow motion upon the following principle : 

Let A, B, C, D represent the numbers of teeth in a train of 
wheels in gear arranged as in the diagram. 
If A = D, and B = C, then A and 
D will rotate with the same velocity in the 
'same direction; but if the equality between 
(A, D) and (B, C) be slightly disturbed, 
we shall produce a small change in the 
value of the train. Suppose, for example, that A is less than D, 




Slow Motion. 24! 

or that A=3i, D=32 ; and, again, that B is less than C, or that 
B = 125, C = 129 : then e, the value of the train, will be 

= AC _ 3_x_i9 ^ 3999 
BD 125 x 32 4000' 

Also, the more nearly the equality is maintained between (A, 
D) and (B, C) respectively, the more nearly will the angular velo- 
cities of A and D be the same, or the more nearly will e be equal 
to unity. 

Thus if B = D =; 100, A = 101, C = 99, 



Let us now arrange A, B, C, D in an epicyclic train, and carry 
back the wheel D so that it shall turn upon the same axis as A. 
The turning of the arm will then set all the wheels in motion 
except A, which is to be made an immovable or dead wheel, and 
we shall have D and A moving relatively to each other just as 
before, that is to say, D will turn very slowly over A at rest. 

FIG. 253. 



DID 

Al 



11 ' I 1 ' i i Miiirn'.niiiyi!! i-i'lil'liMU'Ml " I 1.1 I uri lil'i'i'l 'I'll'' 1 ' 
l-..lK..MMrn:.:".r 

l''l li 



AL an easy example, take wheels of the following numbers 
viz., A = 60, B = 45, C = 40, D = 65. 

AC 60 x 40 12 
- 



-- 



If we now rotate the arm and carry round the train it will be 
found that D makes one revolution when the arm has been carried 
through a little more than 5^ revolutions, which is also evident 

from the formula upon observing that = 5*, which is a little 
greater than 5^. 



242 Elements of Mechanism. 

So, again, taking the formula - = i , and substituting for 
the values given previously, we have in the respective examples, 




a 4000 a 10000 

Hence the arm will make 4000 or 10,000 revolutions respec- 
tively while the wheel D turns round once. 

ART. 195. These examples lead us to compare the move- 
ment of any wheel in an epicyclic train with that in another train 
where the axes are fixed in space, and to regard the subject from 
a different point of view. 

Referring again to the fundamental case, viz., that of three 
equal wheels, A, B, and C, we have seen that if the arm be fixed, 
and A makes one turn, the wheel 
C will also turn once in the same 
direction. But if the arm re- 
volve round A fixed, the wheel 
C will apparently run round just 
as it did upon the last supposi- 
tion, and yet at the end of a revolution of the arm it will be found 
that the wheel C has not turned at all. 

The explanation is that the fixed train gives the absolute motion 
of C due to its connection with A, whereas the epicyclic train 
exhibits the relative motion of C with regard to A, which in this 
case is nothing, because A and C rotate with equal velocities in 
the same direction. 

The same thing is true with respect to any other wheel in the 
train, such as B. Thus, when the axes are fixed in space, A and B 
revolve in opposite directions, and the motion of B relatively to A 
is twice its absolute motion, and thus we account for the fact that 
in the epicyclic train B will rotate twice while the arm goes round 
once. 

So also in Art. 194 the fixed train gives the absolute motion of 
D, viz., fths of a revolution for each revolution of A, and the 
epicyclic train exhibits the relative motion of D as compared with 
thatpf A, viz., -^Vrrth of the movement of A in the fixed train. 
/ART. (96) Another illustration of aggregate motion is found 
-/in Equation clocks. In these nearly obsolete pieces of mechanism 



/> 



Equation Clock. 



243 



FIG. 255. 



the minute hand points to true solar time, and its motion therefore 
consists of the equable motion of the ordinary minute hand plus 
or minus the equation or difference between true and mean solar 
time. 

In clocks of this class the hand pointing to true solar time is 
fixed to the bevel wheel. 

The wheel A moves as the 
minute hand of an ordinary clock; 
the intermediate wheel B is fixed 
to a swinging arm, EB, as in Art. > 
191, and the position of C will * 
be in advance of that of A when 
EB is caused to rotate a little in 
the same direction, and behind 
that of A when EB is moved in 
the opposite direction. 

Thus, as C goes round during 
each hour of the day, the hand 
attached to it may be a few minutes before or behind another 
showing mean time, and deriving its motion at once from A. 

The required motion of EB is obtained from a cam plate, Q, 
curved as in the diagram, and attached to a wheel which revolves 
once in a year. 

ART. 197. In the manufacture of rope the operation of 

FIG. 256. 





' laying,' or twisting the strands 
effected by special machinery. 



into a perfect rope, has been 



244 Elements of Mechanism. 

The Rev. Edmund Cartwright, the inventor of the power- 
loom, was also the first inventor of a machine for making rope. 
The general character of the contrivance will be understood from 
the sketch, which is taken from the specification of the invention. 

The machine itself is called a ' Cordelier,' and consists of a 
frame placed upon a horizontal shaft PQ, and terminating in a 
laying-block R, which serves the double purpose of directing the 
strands to the rollers at K, where they are twisted into rope, and 
of forming a support or bearing for one end of the shaft. 

Three spool frames carry the bobbins, or spools, which contain 
the supply of strands, and the strands, as they are unwound from 
the bobbins, pass through delivery rollers at D, E, and F, and 
thence onward to the laying top. 

All this is simple enough, and might be the invention of any- 
one ; but there is yet a difficulty to be overcome, which we pro- 
ceed to explain. 

Upon examining a rope it will be found that the twist of the 
rope is always in the opposite direction to that of the strands, and 
it follows that it the bobbins were absolutely fixed to the rotating 
frame the strands themselves would be untwisting during the 
whole operation. This untwisting is provided against in a rope- 
walk by the use of two machines, one at each end of the walk. 
The strands are attached to hooks on one of the machines, and 
these hooks are made to rotate with a velocity which exactly neu- 
tralises the twist of the machine which is forming the strands into 
a finished rope. 

Tn the Cordelier the difficulty is at once removed by the intro- 
duction of an epicyclic train. A dead wheel A, so fitted that it 
remains stationary while the shaft PQ rotates within it, gears with 
a second wheel B, and this latter with a third wheel C, equal to 
A, whose axis terminates in one of the spool frames. Now we 
have just proved that in such a train C will run round A without 
rotating at all upon its own axis, and hence the bobbin may be 
carried round without in the slightest degree untwisting the strand. 

In order to make this matter still more apparent we refer the 
student to fig. 257, which is intended to show three positions of a 
spool when rotating in a frame without the intervention of an 
epicyclic train. It is quite evident that the spool has made one 



The Cordelier, 245 

rotation round an imaginary axis through its centre while rotating 
once round the centre of the frame. 

In fig. 258, on the other hand, where an epicyclic train, with 
C equal to A, is interposed, the bobbin will take the positions C, 
C', C", during a revolution, and the rotation just referred to will be 
exactly neutralised. 

FIG. 257. FIG. 258. 




ART. 198. We have stated that the twist of a rope is always 
in the opposite direction to that of the strands, and it may be 
asked, Why is this, and what is the reason that a rope does not 
untwist itself? 

The answer is that any single strand or cord, when twisted up, 
will always tend to untwist in virtue of the elasticity of its fibres, 
and that each separate strand in a rope exerts this tendency 
throughout its whole length ; but since the twist of the rope is in 
the opposite direction, the aggregate of all these comparatively 
feeble forces is felt as a powerful force restraining the whole rope 
from becoming untwisted. 

It follows, therefore, that by putting a little extra twist upon 
the strands of a rope in the process of laying, the rope itself will 
become harder or more tightly twisted. 

If anyone will try and make a small piece of cord out of three 
pieces of string he may at once satisfy himself of the correctness 
of what has been stated. 

Take three pieces of string, or fine sash line, thread them 
through holes in a small plate or disc, to keep them separate, and 
fasten them together at one end, leaving the other ends free. 

Upon twisting the knotted end and slowly advancing the disc, 
a cord will be made which will untwist as soon as it is handled. 



246 Elements of Mechanism. 

Whereas by continually twisting each individual strand, and 
allowing the knotted end to turn in the opposite direction to that 
in which the strands are being twisted, a hard piece of cord may 
be made which will have no tendency whatever to untwist. 

There is a model in the collection belonging to the School of 
Mines which shows this experiment in a striking manner. (Fig. 259.) 

The driving apparatus consists of an arrangement for rotating 
at the same time three hooks. Each hook, /, is formed of a bent 
piece of wire terminating in an upright portion, a, which is 
threaded into a flat disc A. There is a detached sketch of one of 
these bent wires. Upon carrying A round in. a circle, it will be 
found that each hook rotates on its own axis. This motion has 
been explained in Art. 96. 

Take now three pieces of braided sash line, which have no 
twist, and suspend a weight W to each of them. Make a small 
loop at the opposite end of each line, and hang them all up on 
one of the hooks. 

A small conical block B, having a handle H, and grooved as 
in the sketch, is held in such a manner as to receive each line and 
to direct its motion. 

The operator now rotates the hook /, and allows the block B 
to descend slowly while the cord is being twisted. But on looking 
closely at the sash lines it will be found that each weight VV is 
turning on its axis during the whole operation. In truth, the 
weights are made in the form of long cylindrical bars, in order to 
permit this movement. 1 he result is that there is no twist what- 
ever remaining on the individual sash lines or strands. 

Now remove the block, when it will be found that the cord, 
which appears to the eye well made and perfect, will at once un- 
twist, and is, in fact, of no use whatever. It is defective in not 
having any power of retaining the twist which is essential to the 
hardness and durability of a rope or cord. 

The apparatus can, however, be so arranged as to put a strong 
twist upon each individual strand, in such a manner that the twist 
shall be retained in the finished cord, and shall act always to twist 
the same more tightly. 

For this purpose the weights are tied together by a piece of 
string at D, and can no longer ro ate separately. Each strand is 



Twist of a Rope. 



247 



hung on a separate hook, and the respective hooks /, /, /, rotate 
together by the carrying round of the disc A. The block is held 
differently, being placed as near as possible to D, and it is moved 
slowly upwards while the cord is being made. This is exactly the 
operation performed in a rope-walk, except that the strands are 
carried along in a horizontal line. 

FIG. 259. 
a ? a 





There is no difficulty about twisting the cord, for the surplus 
twist put upon the strands causes the weights W to go round to- 
gether underneath the block B, and a well-formed cord is made 
as the block rises. When completed, the string at D may be un- 
tied, the strands may be taken from the hooks, but there is no 
untwisting. On the contrary, the cord will bear handling, and is 
quite hard and durable. 

ART. 199. Many years ago Captain Huddart incorporated 
the invention of the Cordelier into some useful machinery for 
manufacturing rope, and he employed the same epicyclic train, 
but made the wheel C smaller than A in the proportion of 13 to 



248 Elements of Mechanism. 

14, as in the case of the wheel G in Ferguson's paradox. The 
result was that a slight additional twist, or forehard, as it is 
termed, was given to the strands of the rope. 

Among the apparatus belonging to the School of Mines is a 
hand machine for manufacturing fine hard cord, resembling whip- 
cord. It is, in fact, a miniature Cordelier, and instead of the epi- 
cyclic train of wheels for keeping the bobbins parallel during the 
rotation, there is a single dead wheel or grooved pulley, A, and 
FlG 26a three strong india-rubber cords, 

connecting A with the separate 
axes C, C', C", on vhich the 
bobbins are placed. 

Each of the grooved pulleys, 
C, C', C", is of smaller diameter 
than the wheel A, and therefore 
turns slowly backward in the op- 
posite direction to that in which 
it is carried. This fact is made 
clear in the sketch, for the arrow 
indicates the direction of rotation 
of the frame carrying the bobbins, and the dark, lines, aa, bb, cc, 
show the manner in which the respective bobbins rotate back- 
ward. That is, when C arrives at C', the line aa will have turned 
into the position bb, and when it arrives at C", the same line will 
have turned into the position cc. 

Whereas, if the pulleys C, C', C" were each equal to A, the 
line aa would have remained parallel to itself throughout the 
motion. 

In Mr. Smith's wire-rope machine, which is described in the 
papers of the Institute of Mechanical Engineers for the year 1862, 
the bobbins are placed one behind the other in the axis of a re- 
volving fiame, and have simply a slow unwinding motion on their 
axes as the wire strands are run off ; the important result being 
that the rate of manufacture is greatly increased. There is no 
question as to the superiority of this arrangement in a mechanical 
point of view, for the process of laying has no tendency to twist 
the strands when the bobbins themselves lie in the axis of rotation 
of the frame which surrounds them. 




Drilling Machines. 



249 




ART. 200. A further illustration of aggregate motion occurs 
in machinery for drilling and boring. 

In a drilling machine the spindle which carries the cutting 
tool revolves rapidly, and at the same time advances slowly in the 
direction of its length. 

The movement is obtained upon an 'obvious principle, which 
may be stated as follows : 

Conceive a nut, N, to be placed 
upon a screw-bolt, FG, and to be 
so held in a ring or collar that it can 
rotate freely without being capable 
of any other motion. 

If the nut be fixed, and FG be 
turned in the direction of the arrow, 
it is clear that the bolt must ad- 
vance through the nut. If, again, 
the screw be prevented from turning, and the nut be made to ro- 
tate in the same direction as before, the bolt will come back again. 
And, finally, if by any contrivance different amounts of rotation 
be impressed at the same time upon the nut and the screw, the 
bolt will receive the two longitudinal movements simultaneously, 
and the aggregate motion will be the sum or difference of these 
component parts. 

ART. 201. Suppose the wheels D and C to be attached to 
the bolt and nut respec- 
tively, and to be driven by 
the pinions A and B, which 
are fixed upon the same 
spindle ; and let A, B, C, 
D represent the numbers 
of teeth upon the respec- 
tive wheels. 

If (a) be the number 
of rotations made by either 
A or B while the nut fixed 
to C makes m rotations, and the wheel D makes n rotations, 

we shall have- = ^, and - = 4 
a C a D 




2 5O Elements of Mechanism. 

Therefore (a) rotations of A will cause a travel of the bolt FG 
through a space 

fa n) x pitch of the screw 

= a ^? _ M x pitch of the screw. 
ART. 202. We shall proceed to examine the construction of 

FIG. 263. FlG - 2fi 4- 





a small Drilling Machine, which may be worked either by hand 
or by steam-power, but is not self-acting. 



Drilling Machine. 25 1 

The general arrangement of the machine is shown in fig. 264. 
The power is applied to turn the bevel wheel D, which again drives 
C, and causes the case or pipe containing the drill spindle to 
rotate. This provides for one part of the motion, viz., the rotating 
of the drill spindle, and the hand wheel K drives the spur wheels 
M and N, and advances the drill into the work in a manner which 
we shall endeavour to make clear. 

The drill spindle is formed in two pieces, as shown in fig. 263, 
and the upper or screwed portion does not rotate with the lower 
cylindrical portion which carries the drill, but simply moves it up 
and down by means of a collar without interfering with its rota- 
tion. The screwed piece works in a nut forming the boss of the 
wheel N, and is prevented from rotating by a feather sliding in a 
groove or slot which runs along the whole length of the screw, 
and which cannot be seen in the view given in the drawing, the 
feather itself being fixed in a stop-collar at N. 

Hence the rotation of the wheel N, by reason of its connection 
with the hand wheel K, will raise or depress the whole spindle as 
required. 

The rotation of the drill spindle is provided for by cutting a 
groove m n in the lower part of it, and attaching a corresponding 
projection or feather to the inside of the pipe AB. This allows the 
spindle to move lengthways in the pipe, and ensures its rotation 
just as if it were a part of the tube in which it is held. 

A machine of this construction might easily be made self- 
acting, as in a very useful form manufactured by Messrs. Smith, 
Beacock, and Tannett. Here the screwed spindle is not em- 
ployed, but a rack and pinion is substituted for it, and the pinion 
is slowly raised or depressed by an endless screw and worm wheel 
set in motion by a hand wheel similar to K. 

The self-acting portion consists of a small cone pulley, which 
draws off a motion of rotation from the driving shaft, and the axis 
of this pulley is fitted with a second endless screw and worm wheel 
placed just over the hand wheel, and which can be slid into gear 
so as to produce the self-acting motion. 

Thus the same slow rotation may be given to the driving pinion 
on the axis of the hand wheel, by the steam-power, which is other- 
wise given to it directly by the workman ; the cone pulley of course 



252 



Elements of MecJidnism. 



providing for varying amounts of feed according to the require- 
ments of the work. 

ART. 203. A Drilling Machine by Mr. Bodmer, of Man- 
chester, is made self-acting in the following manner : 

The drill spindle (fig. 265) has a screw-thread traced upon it. 
A groove is cut longitudinally along the spindle, and a projection 
upon the interior of the boss of the wheel D fits accurately into 
the groove. 

Thus the spindle can traverse through the wheel D, although 
the spindle and wheel must turn together. 

FIG. 265. FIG. 266. 





A nut H, in the form of a pipe, 
and having a wheel, C, at the bottom 
of it, receives the spindle. This wheel 
and pipe are shown separately in sec- 
tion. 

If a pinion, A, turning in the di 
rection shown by the arrow, engage the wheel D, it will screw the 
spindle rapidly out of the pipe H, and bring it down towards the 
work. 

Suppose a second pinion, B, turning in the same direction as 
A, to act upon C, it will move the nut instead of the screw, and 
the drill spindle will rise rapidly so long as it is prevented from 
rotating. (Fig. 266.) 

Thus far we have provided for bringing the spindle down to 
its work, and for raising it up again. It remains to apply the 
principle of aggregate motion, and to cause the drill spindle to 
become the recipient of these two movements in a nearly equal 



Drilling Machine. 



253 



degree, and thereby to ensure the slow descent accompanied by a 
rapid rotation, which is required in process of drilling. - 

The result of the combination is shown in fig. 267, where the 
wheels A and B are moved together : 
the wheel A tends to depress the spin- 
dle, the wheel B tends to raise it, and, 
since A is greater than B, the spindle 
descends by the difference of these 
motions, having further the motion of 
rotation given by the wheel A. 

The motions of A and B are ob- 
tained from the driving pulleys I, N, 
and L. 

I is an idle pulley, N drives A, and 
L drives B. When the strap is on N 
the drill descends to the work, when 
the strap is on L it ascends from the 
work, and when the strap is partly on 
N and partly on L the drilling pro- 
ceeds. 

The practical objection to this movement is that the rate of 
feed is invariable so long as the train of wheels remains the same. 
It may be thought better to control the feed by means of a cone 
pulley, where the strap can be readily shifted so as to change the 
advance of the cutter. 

ART. 204. A Boring Machine would be employed to give an 
accurate cylindrical form to the interior surface of a steam cylinder. 

In the annexed example the boring FIG. 268. 

cutters are attached to a frame which rides 
upon a massive cast-iron shaft or boring 
bar, and rotates with it : this frame is fur- 
ther the recipient of a slow longitudinal 
movement given by a screw. 

An annular wheel, A, shaped as in the 
diagram, rides loose upon the bar, and 
drives a pinion, P, at the end of the feed- 
ing screw which advances the cutters, the boring bar being recessed 
in order to receive the screw. 





254 



Elements of Mechanism. 



FIG. 269. 



It is quite apparent that as long as the rotation of the wheel 
A is identical with that of the boring bar, the pinion P will not 
turn at all ; and, further, that a slow motion will be impressed upon 

P if the rotation of A be 
made to lag a little behind 
that of the bar. 

A spur wheel, B, is keyed 
to the bar, a small shaft 
fixed at the side carries the 
wheels C and D, and thus 
motion is imparted to A, 
the driver of the feeding 
screw. Let the numbers of 
teeth upon B, C, D be 64, 
36, 35, and let the wheel 
A have 64 teeth, both upon 
the outside and the inside of its circumference, the pitch of the 
screw being \ an inch, and the number of teeth upon the pinion 
being 16. 




5 - > . 
C x A 36 x 64 36 

That is, A loses ^ tn f a revolution for every complete rotation 
of the boring bar. 

At the same time the pinion P moves through J$ x T | or \ of 
a revolution, and the cutter advances through \ ^ x \ an inch or 
through y^th of an inch. 

ART. 205. This slow rotation of the screw which advances 
the boring head may be obtained in a more simple manner by a 
combination which virtually embodies the sun and planet wheels 
of Watt. 

Conceive that two wheels, A and B, of 40 and 80 teeth re- 
spectively, are attached to the bar CAB, which has a centre of 
motion at C. 

If the bar be carried round C, and A be made a dead wheel, 
the effect of depriving A of the rotation due to its connection with 
the arm will be to cause B to rotate relatively to the arm just as if 
the axes of both wheels were fixed in space. 

The movement is shown in the diagram, where A has turned 



Feed Motion. 



255 



through half a right angle from its first position relatively to tht 
arm, while the arm itself has been carried through a right angle. 

The student will distinguish between the absolute and relative 
rotations of B ; the absolute amount of the rotation of B is one 
right angle and a half. 

This also appears from the formula, viz. e = ~^. 



Substituting the values #/=o, e = , we have- 



But (n a} represents the number of rotations of the wheel B 
relatively to the arm while the latter is making (a) revolutions, 
and the analysis therefore shows that the angular velocity of B 
relatively to the arm is half that of the arm itself, and also that 
both rotations take place in the same direction. 

Further, it must be noted that the position of C makes no 
difference in the result, which will be the same if the point C be 
somewhere between A and B. 

1 





In the application of this movement to the boring machine, 
the centre of motion is between the axes of the wheels, in the 
line marked C in the diagram, and the numerical value of e is 
less than ^, probably about ^. 

The wheel B is placed upon the axis of the screw which 
advances the boring cutters, the rotating arm being now a part 
of the solid end of the boring bar ; the wheel A rides upon a 
separate stud, and is attached to a bar AD of some convenient 



2 5 6 



Elements of Median ism. 



length which passes through and rests upon a fork in an indepen- 
dent upright support placed at some little distance from the 
machine. 

As the wheel A is carried round the axis of the boring bar 
this rod slides a little to and fro in the fork, and controls the 
wheel A so as to render it impossible for it to rotate, or, in other 
words, to make it a dead wheel. 

The wheel B will now turn slowly under the action of A so 
far as its position relatively to the boring bar is concerned, and 
upon our supposition, the screw will advance the boring cutters 
by a space equal to its pitch in five complete revolutions. 

This would give a feed dependent upon the pitch of the screw, 
which could of course be varied at once by changing the wheels 
A and B. 

r. 206. Sir J. Whitworth's Friction Drilling Machine is an 
application of the principle of aggregate motion. 
AD is the drill spindle, which is driven in the usual manner by 
the bevel wheel B. 

E and F are two worm wheels embracing the screwed portion 
of the spindle upon opposite sides. They 
are of peculiar construction, being hollowed 
out so as to fit against the small screwed 
spindle, and they work with a V-threaded 
screw upon AD. 

If E and F be prevented from turning, 
they will form a nut through which the spindle 
will screw itself rapidly. 

If E and F be allowed to turn quite freely, 
the drill spindle will set them in motion, and 
the nut will be virtually eliminated. The drill 
spindle may then be regarded as the recipient 
of two equal and opposite motions : it is 
depressed by screwing through the nut, it is 
elevated by the turning of the wheels. 

If the rotation of the wheels be in any 
degree checked by the application of friction, 
the equality is destroyed, and the drill spindle 
descends to a corresponding extent. 



FIG. 271. 




Watt's Indicator. 



257 



FIG. 272. 



A friction brake, regulated by a screw, restrains the motion of 
E and F, and gives a perfect command over the working of the 
machine. 

When B is at rest the worm wheels act upon the screwed part 
of the spindle just as a pinion does upon a rack, and the drill can 
be rapidly brought down to the work. 

This method of converting a screw and worm wheel into a rack 
and pinion is quite worthy of attentive consideration : it is em- 
ployed in the well-known lathes by the same firm. 

ART. 207. Waffs Indicator is an instrument used to ascer- 
tain the actual horse-power of a working steam-engine. The 
principle upon which it is constructed is the following : 

A pencil oscillates through 
the space of a few inches in a 
horizontal line, with a velocity 
which always bears a fixed ratio 
to that of the piston, whereby its 
motion is an exact counterpart 
upon a very reduced scale of the 
actual motion of the piston in the 
steam cylinder ; and at the same 
time it is the subject of a second 
movement in a vertical line, which 
is caused by the pressure of the 
steam or uncondensed vapour in 
the cylinder, and occurs whenever 
the pressure of the steam or va- 
pour upon one and the same side 
of the piston of the engine be- 
comes greater or less than that of 
the atmosphere. 

Under the influence of these 
independent motions the aggre- 
gate path of the pencil will be a 
curve which is capable of inter- 
pretation, and which affords a 
wonderful insight into actions 
which are taking place in the interior of the cylinder. 
S 




2 5 8 



Elements of Mechanism. 



An excellent early form of the apparatus is known as 
McNaught's indicator, and consists of a small cylinder, A, fitted 
with a steam-tight piston, B. The piston rod, BD, is attached to 
a spiral steel spring, which is capable of extension and compression 
within definite limits, and is enclosed in the upper part of a tube 
which carries the cylinder A. 

The pencil is attached to a point in the rod BD, and traces 
the indicator diagram upon a piece of paper wrapped round a 
second cylinder by the side of the first. 

The cylinder, A, is freely open to the atmosphere at the top, 
and a stopcock admits the steam from below when required. The 
indicator is usually fixed upon the cover at one end of the steam 
cylinder of the engine. When the stopcock is opened and the 
lower side of B is in free communication with the interior of 
the cylinder, the pressure of the steam will be usually greater or 
less than that of the atmosphere : if it be greater, B will rise 
against the pressure of the spring, and if it be less, the pressure 
of the atmosphere upon the upper surface of B will overcome the 
resistance of the spring and cause the pencil to descend. 

FIG. 273. 




At the same time, the cylinder which carries the paper is made 
to turn with a motion derived at once from that of the piston in 
the engine, but much less in degree, and thus a curve is traced 
out somewhat of the character represented above. 



The Indicator. 259 

Here PQ is the atmospheric line, and is the path of the 
pencil when the pressure of the steam is equal to that of the 
atmosphere, or when the spiral spring is neither extended nor 
compressed. 

As the steam enters the cylinder, the piston may be supposed 
to be descending, and the pencil to be describing the upper por- 
tion of the curve : when the piston returns, the pencil moves to 
the left through DEA, and thus the diagram is traced out. We 
may examine this matter with more particularity as follows : the 
steam is admitted when the piston reaches the top of its stroke, 
and the pencil rises with a rapid motion from A to B ; the full 
pressure of the steam is then maintained while the pencil, recording 
a portion of the travel of the piston, moves from B to C ; at C the 
steam is cut off, and the pencil falls gradually as the steam expands 
with a diminishing pressure ; at D the steam pours into the con- 
denser, and the fall becomes sudden ; from E to A the cylinder is 
in full communication with the condenser, and the pencil describes 
a line somewhat inclined to the line PQ, the position and form of 
which depend upon the perfection of the vacuum in the condenser. 

The strength of the spiral spring being ascertained, the curve 
tells us exactly the number of pounds by which the pressure of 
the steam urges the piston onward during every inch of its path 
in one direction, and the amount of resistance which the uncon- 
densed vapour or gases existing in the condenser oppose to its 
passage in the other direction. The area of the curve, therefore, 
affords an estimate of the work done in the engine during one 
complete stroke, and is a graphic representation of the same. The 
engineer estimates this area by simple measurement in the most 
direct manner which occurs to him, and the actual indicated 
horse-power is obtained by multiplying the work done in one 
stroke by the number of strokes made in a minute, and then 
dividing by 33,000, the number of foot-pounds which form the 
measure of rate of work called a horse-power. 

ART. 208. The object of the indicator being to ascertain the 
xact pressure of the steam or vapour in the cylinder at each 
point of the stroke of the piston, it has been found to be a great 
advantage to diminish as much as possible the play of the spring 
>vhich controls the pencil. In this way the vibration and irregu- 



260 



Elements of Mechanism. 



larity of motion of the pencil is greatly reduced. But the play 
given to the spring determines the height of the diagram, and we 
do not wish to reduce this, but rather the contrary. It is not easy 
to reconcile these contradictory requirements, but, nevertheless, a 
form of indicator has been invented by Mr. Richards which solves 
the difficulty, and has become most deservedly popular. 

It is an ingenious application of the combination of two bars 
and a link forming a parallel motion, and will be understood at 
once from the drawing, which is taken from a small model repre- 
senting very closely the essential parts of an actual instrument. 

The parallel motion bars AB and CD carry the pencil, which 
traces out upon a drum a copy of the vertical movement of the 
piston E of the indicator, but magnified by reason of the attach- 
ment of the piston to a point S near the fulcrum of the bar CD. 

FIG. 274. 




The principle of the apparatus is precisely the same as that which 
we have already explained, and the only difference consists in the 
application of the parallel motion bars to enlarge the diagram. 



The Indicator. 261 

The drum derives its motion from any part of the engine 
whose movement is coincident with that of the piston, and the 
spiral spring can be changed so as to suit different engines. The 
connecting link is not set perpendicularly to the bars AB, CD, but 
makes an angle with them as shown, an artifice which causes the 
pencil to describe a line free from any sensible curvature. 

The parallel motion is set out in a separate diagram, in order 
that it may be thoroughly understood. 

FIG. 275. 




The link RS is parallel to BD when the motion begins, and 
it remains parallel throughout, for R and P are both constrained 
to describe vertical straight lines. 

Hence we have the pantograph in a disguised form. 
Also, travel of P : travel of R = CD : CS. 

In the indicator as constructed the movement of R is mag- 
nified about four times. 

It should be understood that the frame carrying the motion 
bars is attached to a collar which can be rotated on the cylinder, 
whereby the pencil is readily brought up to the paper or removed 
from it. 

ART. 209. There is a curious movement derived from the 
employment of a dead wheel in a train which has been applied by 
Mr. Goodall in a machine used for grinding glass into powder by 
the action of a pestle and mortar. The pestle is made to sweep 
round in a series of nearly circular curves contracting to nothing, 
and then expanding again so as to command the whole surface 
of the mortar. 

We shall show that the contrivance is merely a solution of the 
problem of obtaining an expanding and contracting crank. 

Let CQE be a crank whose centre of motion is D; conceive 



262 



Elements of MecJianism. 



that D is a dead wheel on the same axis, and that A is a larger 
wheel riding upon one end C of the crank arm. 

FIG. 276. 




Suppose, further, that a piece Q, capable of sliding along CE, 
is attached by a link SQ to a point S in the circle A, which is not 
its centre ; and, finally, that a pencil at P is connected with Q by 
a link PRQ constrained always to pass through a fixed point R. 

As the circle A and the crank CQ travel together round the 
dead wheel D, it has been proved in Art. 195 that the wheel A 
will turn relatively to the arm just as it would do in an ordinary 
train with fixed axes. Hence the point S travels slowly round in 
the dotted circle, thereby causing the point Q to move to and fro 
along CQ. 

It may be arranged that Q shall start from D, and it will travil 
along CE through a space equal to twice CS. When Q is at D, 
the point P is motionless, whereas, while Q moves further from D, 
and continually sweeps round, by virtue of its being a point in a 
revolving crank, it is evident that P will trace out an expanding 
spiral, which will return again to nothing when Q is pulled back to 
D by the action of the wheel A. 

It now only remains for us to consider what would be the 
actual construction of the apparatus. 

The drawing is taken from a model, and not from the machine 
itself. 

The crank CDE is a bar whose parallel arms are connected by 
a vertical piece, and which carries the wheel C upon one arm and 
the sliding piece KQ upon the other arm. This crank is driven 
by the spur wheels L and M connected with the handle H; it 



Expanding Crank. 



263 



therefore rotates round the axis of the dead wheel D, and carries 
C and KQ upon opposite sides of the vertical axis through D. 




The link SQ connects the wheel C with the piece KQ, and 
this latter piece is again connected by QR with the pestle, it being 
provided that QR shall pass through a guide at some fixed point 
about half-way between Q and the mortar. The pestle is swung 
from a ball and socket joint at some convenient height above P. 

The rotation of the crank round the dead wheel causes C to 
turn slowly upon its own axis, the point S therefore travels slowly 
round C, hence the end Q of the connecting rod QRP is some- 
times at a distance from D, and at other times is exactly over it, and 
during the whole time Q is a part of the crank DQ, and sweeps 
rougd with the arm. Thus the required motion is provided for. 
./ >^ART. 210. The oval chuck affords an instance of aggregate 
Y/ motion. It is based upon the following property of an ellipse, 
/ which is taken advantage of in constructing elliptic compasses for 
drawing the curve. 

Let ACA', BCB, repre- 
sent two grooves at right 
angles to each other, and 
traced upon a plane surface ; 
PDE, a rod furnished with 
pins at D and E. If this rod 
be moved into every possible 
position which it can assume 
while the pins remain in the 
grooves, the point P will de- 
scribe an ellipse. 




264 



Elements of Mechanism. 



Draw PN perpendicular to AC, and PM perpendicular to CB'. 
Let CN = * PE=*, 



PM , y PN EM 



. * 2 , y^PM 2 + EM 2 = 
' * a* 2 ~~ PE 2 
which is the equation to an ellipse. 

In drawing an ellipse we should fix the paper and move the 
rod over it, but in turning an ellipse in a lathe we should fix the 
describing tool and move the piece of wood or metal underneath 
it ; thus the conditions of the problem become changed, and the 
construction is modified accordingly. 

An equivalent for the grooves ACA', BCB' may be arrived at 
as follows : 

Describe a circle about E of radius larger, than ED, and let 
two parallel bars, QR, ST, be connected by a perpendicular 
link HK, equal in length to the diameter of the circle, and thus 
form a rigid frame embracing the circle, and capable of moving 
round it. 

FIG. 279. FIG. 280. 

B 





As the frame moves round the circle we must provide that 
HK shall pass through D in every position as represented in the 
diagram. 

If we draw DC parallel to QR, and EC parallel to HK, 
it is easy to understand that the imaginary triangle DCE in fig. 
280 is exactly the same as the triangle DCE in fig. 278 and exists 



The Oval Chuck, 265 

throughout the motion ; and that whereas we formerly moved the 
bar EP over a fixed plane and described an ellipse, so now we 
have arranged to obtain the same motion with a fixed bar and a 
movable plane, and shall trace out precisely the same curve. 

This is a very good example of aggregate motion. The plane 
upon which the ellipse is traced is the subject of two simultaneous 
movements : by one of them a line, HK, in the plane is made to 
revolve round D as a centre, and by the other the same line re- 
ceives a sliding motion in alternate directions through D. 

Thus an oval, or more properly an ellipse, may be turned in 
the lathe. 



266 Elements of Mechanism. 



CHAPTER VIII. 

ON TRUTH OF SURFACE AND THE POWER OF MEASUREMENT. 

ART. 211. The subject matter comprised under the title of 
this chapter is so large that it cannot be discussed fully, and all 
that can be done is to present a brief sketch of some important 
facts connected with it. 

We have to describe a method of mechanical measurement, 
founded upon truth of surface, which is probably the reverse 
of that which most persons would form for themselves. The 
idea of measurement is commonly associated with optical con- 
trivances, whereby, if we desired to measure some minute interval 
of length, say in ten-thousandths of an inch, we should naturally 
proceed to the task armed with powerful lenses or microscopes, 
and relying mainly on the sense of sight. It would be something 
quite novel and unexpected to discover that the sense of touch 
would do more for us than the eye, and that, in mechanical 
measurement, it is more easy to feel minute differences of size by 
the aid of surfaces properly prepared and adjusted, than it is to 
recognise and compare such differences in the field of view of a 
microscope. 

We shall presently refer to a measuring machine, used in the 
construction of difference gauges, wherewith a workman can 
readily test gradations of size differing by t uuooth of an inch, 
and may in special cases carry on the operation as far as 4u<Wo tn 

Or 7nTOTTO th f an inch - 

Inasmuch as a machine of this kind is a piece of constructive 
work perfected by the aid of other machines, such as the lathe 
and the planing machine, in which truth of surface is all import- 
ant, it will be useful to consider in the first instance the method 
of originating a plane metallic surface. 

Every one knows that a plane surface is an ideal thing, which 



A Surface Plate. 267 

the geometrician arrives at by an operation of the mind, but 
which has no real existence. 

In the year 1840 Sir J. Whitworth brought to the notice of 
engineers a new method of preparing metallic surfaces, and he 
submitted specimens of cast-iron plates so prepared, which he 
called true planes. 

A ' true plane ' is commonly spoken of in the workshop as a 
'surface plate,' and is made of cast iron, being ribbed at the back 
and resting on three points of support. The truth which it pos- 
sesses is of course approximate, and its surface, when carefully 
examined, will be found to consist of a vast assemblage of minute 
bearing faces which lie very nearly in the same geometrical plane, 
the object aimed at being to distribute these bearing faces as 
nearly as possible at equal distances from each other. 

The preparation of a standard surface plate, and the method 
of employing it for the multiplication of other identical plane 
surfaces, has been a distinct invention in mechanics, the importance 
of which can scarcely be over-estimated, and we proceed to give 
an account of it. 

ART. 212. Up to the year 1840, the process relied upon for 
obtaining plane surfaces on metal plates, and indeed the only one 
practically used, had involved the operation of grinding two plates 
together with emery powder and water. While the operation was 
going on the plates under preparation were occasionally compared 
with a standard plate or plane, but the standard plate was imper- 
fect, the method of comparison was uncertain, and the smooth 
surface given by the process of grinding was entirely deceptive 
if regarded as an evidence of truth. 

The operation of grinding fails for the obvious reason that 
the action of the powder cannot be restricted to those parts only 
which are in error, and it is clear that a complete control over 
the successive removal of any portions of the surface which are 
believed to be out of truth is the first thing to be sought for. 
Such a control can be obtained by the use of a scraping tool, 
which, acting like one tooth of a file, is competent continually to 
remove portions of the metal in the form of a fine powder or 
dust, and it will be shown that by a systematic method of com- 
parison it becomes possible to produce planes which rival in 



268 



Elements of Mechanism. 



accuracy, though not in polish, the brilliant mirror presented by 
the surface of mercury when in absolute repose and undisturbed 
by any ripples. 

A surface plate is always made of cast iron, and has usually 
taken the form of a rectangular plane table, ribbed at the back, 
and resting upon three supports or bearings. 

Three supports are necessary in order that the plate may 
remain under like conditions wherever it is placed. If it were 
supported on four legs, and the foundation gave way under any 
one leg, the plate would become distorted and untrue, but with 
three legs it must take an equal bearing under all circumstances. 
It may be thrown out of level but it will not be distorted. The 
same rule applies in lifting a plate : it must be hung from three 
points. 

The form to be assigned to the plate becomes material when 
the question of lifting or supporting it comes under consideration. 
In order to avoid distortion, a surface plate ought to press equally 
on its three supports, or be lifted by equal tensions, and for this 
purpose its centre of gravity should coincide with that of the 
triangle formed by joining the points of support or of suspension. 
Inasmuch as a rectangular plate of uniform thickness has no 
symmetry of form with regard to a triangle, it follows that 
some polygonal form is theoretically preferable, and accordingly 




a hexagonal plate, such as that shown in the diagram, has been 
adopted in later years by Sir J. Whitworth. The drawing repre- 



Preparation of a Surface Plate. 269 

sents a surface plate in plan and elevation, the circular ribs em- 
ployed for strengthening the plane table being apparent, as well 
as the legs or supports. 

A tripod frame binds the legs together, and the plate can be 
slung by a handle screwed into the tripod, whereby it is subject 
to the same mechanical conditions whether it is resting on the 
supports or is suspended by the handle. 

Having decided on the form of the plate and the method of 
supporting it, we have next to describe the operation of obtaining 
a plane surface. 

ART. 213. A number of good castings having been made, 
three plates are prepared in the planing machine by levelling the 
bosses or supports and by planing over the surfaces. 

The first step is to ascertain the most perfect plate of the 
three chosen, which is done by laying a good straight edge upon 
each surface, and noting that which agrees most nearly with it. 
[A straight edge is a flat steel bar, on the thin edge of which a 
plane surface has been formed. In practice the surface of a 
straight edge is a true plane, but we shall consider it as less 
perfect than the plane we are about to originate.] Let us call the 
three selected plates by the letters A, B, C, and let A be chosen 
as the primary model. The operation of scraping is now relied 
upon for the continual correction and improvement of the sur- 
faces. 




A scraping tool, as shown in elevation and plan in fig. 282, is 
forged from a bar of steel, the cutting edge being formed by 
grinding two facets at an angle of about 60. The edge is care- 
fully set on an oil-stone, and for finishing tools the angle of the 
scraper may be increased up to 120. 

The workman holds the tool in his right hand, pressing the 
edge on the surface to be scraped with his left hand, and moves 
the tool to and fro through a small space, thus taking off very 



270 Elements of Mechanism. 

small quantities of metal in the form either of minute shavings 
or of fine powder according to the degree of force exerted. 

It has been stated that scraping is a very delicate application 
of the principle of filing, the edge of the scraper acting as if it 
were one tooth of a very fine file, the object being to detach from 
any portion of the surface as much or as little of the metal as 
may be desired, and to confine the operation to the precise spot 
which may be in error. 

FIG. 283. 




In order to present some idea of the result of scraping on 
the appearance of a surface, the annexed drawing has been en- 
graved from a photograph, and shows the peculiar mottled ap- 
pearance which is due to the multiplied action of the scraping 
edge in every direction as the work advances. 

ART. 214. We have now to endeavour to arrive at a perfect 
coincidence between three surfaces, A, B, and C. 

It is a geometrical fact that if three surfaces can be brought 
into exact coincidence when compared and interchanged in every 
position, each of them must be a plane surface. Thus three 
standard planes must be produced in the endeavour to form one 
only. 

Taking two of the plates, as A and B, the method is to smear 
the face of A very lightly with red ochre and oil : the plate B is 
then lowered upon A, so that the colouring matter may adhere to 
the surface of B wherever there is contact. B is then scraped 
at all those points to which the colouring matter has adhered, A 
in its turn being wiped clean, and B being coated with the mix- 



Preparation of a Surface Plate. 27 1 

ture, after which the operation is reversed, and A is submitted to 
the scraper. The process is continued until the contact between 
A and B is made as perfect as possible, the bearing points being 
so numerous and evenly distributed that the entire surface ap- 
pears reddened. But although A and B are in perfect contact 
throughout, it may be that B is convex and A concave, as in 
fig. 284. Take now the third plate C, and bring it into perfect 
coincidence with B by scraping, and then compare A and C 
together as in the lower diagram, when the error, if any, will be 
manifest by the failure of absolute contact as indicated by the 
colouring matter. 

FIG. 284. 



A 



To bring A and C nearer the truth, equal quantities must be 
scraped away from both surfaces at the points of contact. When 
this has been done with all the skill the mechanic may possess, 
and A and C are brought into coincidence with each other, as in 
the sketch, the next step is to bring B up to both. The art here 
lies in getting B between A and C in the probable direction of 
the true plane. 

Taking now B as an improved standard for comparison with 
A and C, the process commences de novo y and is carried on in a 
regular series of comparisons, which result in a gradual approach 
towards absolute truth. At length the inherent imperfections of 
the material and of the tools render it impossible to proceed 
further, and the most watchful care is necessary in order to guard 
against the introduction of fresh errors ; the penalty for scraping 
off the slightest excess from any part being the performance of 
the difficult task of lowering the entire surface to the same extent. 

ART. 215. The conditions which a surface plate should fulfil 
are the following : 



272 Elements of Mechanism. 

1. The bearing surfaces should all lie in the same plane. 

2. They should be distributed as nearly as possible at equal 
distances from each other. 

3. They should be sufficiently numerous for the particular 
application intended. 

When a surface plate is completed the bearing faces will be 
found to have acquired a degree of polish, from continual rub- 
bing, and they will appear as numerous bright spots dotted over 
the plane. The plate itself has not the appearance or surface of 
a polished mirror, but it possesses, nevertheless, a considerable 
power of reflecting light, and might, for example, be used as a 
reflector for throwing on one side an image of the carbon points 
of an electric lamp. 

If two well-finished surface plates be wiped with a dry cloth 
and laid one upon the other, the upper plate will appear to float 
and become buoyant, as if some lubricating matter existed between 
them. If the upper plate be somewhat raised and allowed to fall, 
there will be no metallic ring, but the blow emits a peculiar 
muffled sound. These effects have been attributed to the presence 
of a cushion of air. If, again, one plate be carefully slid over 
the other so as to exclude the air, and then pressed together, it 
will be found extremely difficult to separate them. Here again 
it has been supposed that the plates adhered by atmospheric 
pressure. 

Dr. Tyndall has upset this hypothesis as to the adherence of 
two surface plates by atmospheric pressure, and has described his 
experiments in a lecture given at the Royal Institution. 

Having obtained a contact between two small hexagonal 
Whitworth plates, he found that the plates remained adherent in 
the best vacuum obtainable by a good air-pump. The vacuum 
was still further improved by filling the receiver with carbonic 
acid, and absorbing the residue with caustic potash. In this 
way the atmosphere was reduced until its total pressure on the 
surface of the hexagon amounted to only half a pound. The 
lower plate weighed three pounds, and to it was attached a mass 
of lead weighing twelve pounds. Though the pull of gravity was 
thirty times the pressure of the atmosphere there was an excess 
force, and the weight was supported. 



Rectangular Bars. 



273 



It is also noticed that the amount of vacuum formed in the 
receiver of an ordinary air-pump has little or no power in dimin- 
ishing the floating effect which is observed when one plate lies 
upon the other. The floating is just as apparent under the 
exhausted receiver as in the open air. 

The conclusion at which Dr. Tyndall arrives is that the plates 
adhere by the molecular attraction of the bearing points brought 
into close contact by reason of the near approach to absolute 
truth of surface. 

ART. 216. We pass on to describe the method of producing 
a rectangular bar with plane sides and plane ends, which may be 
taken as the type of bars hereafter to be used in the measuring 
machine. 

Fiu. 285. 





A righft-angled groove with plane faces is prepared and made 
as true as possible by straight edges and squares. A rectangular 
bar, fitting the groove as shown in fig. 285, is also prepared and 
rendered approximately true. The bar is then laid in the groove, 
and the faces of the groove are brought by scraping into coinci- 
dence with those of the bar. The bar is next rotated through 
^th of a revolution, so that the angle marked (i) takes the position 
of that marked as (2) in the diagram, and the surfaces are again 
compared and corrected. This goes on till a coincidence is ob- 



274 Elements of Mechanism. 

tained in every position, when the bar and groove must both have 
their surfaces at right angles. 

The geometrical principle is that the four interior angles of a 
quadrilateral figure are equal to four right angles, and if these 
angles be equal, each must be a right angle. 

Having obtained a bar with plane sides, it remains to work up 
the ends into true planes at right angles to the axis of the bar. 
For this purpose one end of the grooved rectangular block 
ABCDE is made plane, and as nearly as possible perpendicular 
to the axis or edge of the groove. The bar is laid in the groove, 
and a trial plane F is brought into contact with the plane end of 
the block and with the end of the bar. The latter surface is then 
made a true plane, and occupies the position i 2 3 4 in the diagram. 
The bar is now rotated through half a revolution, so as to take 
the position 3412, and we have a ready method of testing whether 
the plane end is or is not perpendicular to the axis of the bar. 
If the line CBD, fig. 286, be nearly but not quite at right 
angles to AB, and if the system ABCD be 
rotated through 180 about AB as an axis, 
the lines BC and BD will take the respective 
positions BC' and BD', and the true perpen- 
dicular to AB will bisect each of the angles 
CBD' or C'BD. In other words, the true 
perpendicular lies in HK, which is half way 
between the two positions occupied by CD. 

Applying our proposition to test the di- 
rection of the plane end of the bar, we shall 
probably find that one of the angles of the end-face is alone in 
contact with the trial plane after the rotation. This shows that 
both the plane ABEDC and the end of the bar are out of truth, 
and an equal quantity must be taken from each, the directions of 
the plane surfaces being shifted through half the supposed angle 
of error. The operation is laborious, as it involves an alteration 
in the direction of the whole surface of a plane. 

Having brought the trial plane into perfect coincidence with 
the end of the bar when the angles (i) and (3) are successively 
uppermost, the same thing has to be repeated with the angles (2) 
and (4), and if the coincidence be perfect in every position during 




End Measure. 275 

the rotation, it follows that the plane end of the bar must be truly 
perpendicular to its axis. 

The final operation is to remove the bar from the groove, and 
rub it lightly on a surface plate. 

ART. 217. Every mechanic is familiar with measurement by 
callipers, whereby to judge of differences of size by the degree 
of tightness which is felt when pieces of metal are passed between 
the legs of the instrument. Supposing it were required to com- 
pare two spindles, for the purpose of making one an exact copy of 
the other. Calling the original A, and the copy B, the workman 
would adjust the callipers until he had obtained a certain degree 
of tightness when passing them over the surface of A. He would 
then gradually reduce B, until, to his sense of touch, the feeling 
of tightness of the instrument was alike when passing it over A 
or B indifferently. 

The comparison is not easy, for there are many sources of 
error. The ends of the callipers are blunt edges, imperfectly 
shaped, and it is difficult to hold the legs of the callipers in a 
plane exactly perpendicular to the axis of the cylinder which is 
being tested. The transverse section of a cylinder is a circle, 
whose diameter gives the measure of the cylinder, but an oblique 
section of a cylinder is an ellipse, whose major axis is greater than 
the diameter of the cylinder. When the callipers are so held as 
to embrace a transverse section the observation is correct ; whereas 
when they embrace the opposite arcs of an oblique section the 
observation is worthless. Although a skilful workman may use 
callipers so as to obtain very good results, it is evident that he 
struggles against an inherent defect which cannot be remedied. 

In like manner callipers have been constantly employed for 
the conversion of line into end measure. In effecting this inter- 
change the workman lays his callipers upon the graduations of a 
foot rule, and reads off the interval as nearly as he can. He then 
shapes some piece of metal which is to be made of given dimen- 
sions until the callipers pass over it with an estimated degree of 
tightness, and he thus transfers an interval on the rule to some 
part of a solid body. 

There are three difficulties in the way, and one is very serious. 

i. It is impossible to determine how nearly the end of each 



276 Elements of Mechanism. 

leg of the callipers coincides with the centre of the corresponding 
line of graduation of the rule. 

2. There is the difficulty of holding the instrument in a plane 
perpendicular to the surfaces under measurement. 

3. It is uncertain to what extent the legs of the instrument 
may spring or yield under the variable pressure of contact. 

Some years ago a standard inch was unknown in the work- 
shop, and the graduated foot-rules were often incorrectly divided. 

The subject of mechanical measurement remained in a state 
of uncertainty and confusion until Sir J. Whitworth applied himself 
to the construction of end measure standard bars, as well as of a 
machine for end measuring. 

Before commencing that task he had satisfied himself that the 
only practicable mode of measurement suited for the workshop 
should be one founded on truth of surface and the sense of touch 
the delicacy of the nerves of feeling being, in fact, a thing quite 
disregarded and neglected by all others who had applied them- 
selves to improving mechanical measurement. He undertook 
the novel task 'of showing how to measure minute differences 
of size which were simply felt, and not observed or recognised by 
the eye, and he constructed a machine for reproducing in a much 
more perfect manner the ordinary callipers of the workshop. 

Conceive now that the ends of the legs of the callipers are 
replaced by two parallel true planes fixed at a distance equal to 
the diameter of the cylinder, and that these planes can be caused 
to approach or separate by almost insensible intervals. We have 
then a measuring machine of the highest delicacy and precision. 
The sense of touch can be tested in a manner that will surprise 
any one who approaches the subject without previous training or 
experience, and differences of ten thousandths of an inch become 
palpably evident. Taking the workshop measuring machine as 
constructed by Sir J. Whitworth, we find that its general appear- 
ance is that of a small turning lathe. 

i. There is a cast-iron bed formed of two parallel cheeks, 
connected at the ends, and stiffened by intermediate ribs, the 
bed itself being supported upon standards or feet placed at the 
respective ends. These feet consist of narrow longitudinal fillets, 
two being placed at one end of the frame and one at the other, 



Measuring MacJiine. 



27; 




278 Elements of Meclianism. 

so that the machine rests on three supports, just as if it were an 
ordinary surface plate. 

2. The upper part of each cheek is spread out into a flange 
having one inclined and one vertical face, so as to form together 
a guide for the headstocks B and C, whereof C is fixed and B is 
capable of being traversed along the bed by means of a hand- 
wheel F attached to a quick threaded screw which lies between 
the two cheeks of the bed, and is supported on cylindrical bearings 
at each end. 

The guiding faces of the flanges are carefully scraped and 
made true planes, as are also the surfaces of the headstocks which 
rest upon them ; and since the intersection of two planes is a 
straight line, it follows that the headstock B traverses in a definite 
direction along the plane bed of the machine, and that any point 
in it describes a close mechanical approximation to an exact 
straight line. 

3. A hole is bored from end to end of the upper part of each 
headstock, great care being taken to make it truly cylindrical, and 
with its axis parallel to the bed of the machine. Also the two 
cylindrical holes have a common axis. 

Each bore is then fitted with a cylindrical sliding piece, the 
diameter of the sliding piece being slightly less than that of the 
internal surface according to a difference previously determined 
and tested by difference gauges. The method of doing this will 
be explained hereafter. 

Each cylindrical sliding piece is prevented from turning round 
in the headstock by a narrow key which forms a feather, and is 
recessed, partly into a longitudinal groove cut along the under 
side of the sliding piece and partly into the headstock itself. The 
cylinders terminate in the true planes D and E, which are circular 
in form, and as nearly as possible perpendicular to the axes of the 
respective cylinders. 

4. The measuring plane D is movable by a screw with a 
graduated hand-wheel, the object of which is to put the plane 
approximately into any required position. 

The actual measurement is performed on the side of the 
headstock C. Here the sliding cylinder which terminates in the 
true plane E is adjusted by a screw with twenty threads to the 



Standard Gauges. 279 

inch, and terminating in the hand-wheel P, having 500 divisions 
on its rim. A pointer R registers the motion of the wheel, and it 
follows that a movement of the rim of the wheel through the 
space of one graduation traverses the plane E through y^^th 
of an inch. 

Since the large wheel P is 1 1 '8 inches in diameter, and has 
500 graduations, it follows that the space travelled over in passing 
from one graduation in the rim to the next in order is 741 times 
that through which the measuring plane is shifted. In other 
words, if the measuring plane advances by T ^o7>th f an mcn > 
a graduation in the rim of the large wheel advances by ^^-^ths 
of an inch, which is rather more than ^th of an inch, an interval 
that can be readily observed without assisting the eye by lenses. 

In this machine there is space for measuring circular gauges 
up to 6 inches in diameter, and bars up to 12 inches in length. 
But for ordinary purposes, and for the construction of small 
gauges, a much smaller instrument will suffice, the working parts, 
however, being the same. 

ART. 218. We have now to speak of the construction of 
case-hardened cylindrical gauges, or standards, for comparison in 
the workshop, and the drawing shows the two principal forms. 




1. There is an external gauge A, cylindrical in form, with a 
handle, and of varying diameters from T \j- inch up to 2 inches. 

2. There is an internal gauge B, also cylindrical, and exactly 
fitting upon A. 

These gauges are made in pairs, and serve to test the diameter 



280 Elements of Mechanism. 

of any solid cylinder, or of any cylindrical opening having the 
same diameter. 

In the collection of the School of Mines are two of these ex- 
ternal or plug gauges, and one internal or collar gauge, whereof 
the respective diameters are one inch, one inch less T ^o^o tn mcn > 
and one inch. 

Calling them in the order as above stated by the letters A, 
A', and B, we find these results : 

1. Wipe A and B, which are exact i-inch gauges, with a clean 
dry cloth, and endeavour to pass one over the other. We cannot 
do it, for the surfaces at once bite together, and we should say 
that the plug was too large for the collar. 

Now rub a very little of the finest oil upon the surfaces of A 
and B, and it will be found quite easy to pass B upon A. If A 
be held in a vertical position, B will slowly sink from the top to 
the bottom of the cylinder. The smoothness and yet tightness of 
the fit is most remarkable, the oil preventing the adhesion and 
jamming together of the metallic surfaces. 

2. We next proceed to test A' and B, remembering that the 
diameter of A' is y^g-uth of an inch less than the internal dia- 
meter of B, and we begin by wiping the surfaces so that all oil is 
removed, and they are perfectly dry and clean. 

It will now be found that B fits quite loosely on A'. If the 
gauge A' be held in a vertical position, B will fall freely from the 
top to the bottom. 

Again rub some oil upon the surfaces, which will fill up the 
vacant space, and A' will pass through B very smoothly, but some 
slight resistance will be felt. 

By handling these gauges the difference of fit due to a difference 
of TTf^rth of an inch becomes very apparent. 

The practical value of difference gauges is well understood. 
For instance, the cylinder of the moving headstock of a lathe 
requires as good a fit as possible, but that means that a true and 
proper allowance should be made in the size of the parts working 
together, and Sir J. Whitworth states that in the case of an ordinary 
lathe the hole in the headstock should be ^oW 11 of an mch larger 
than the cylinder. 

In like manner gauges would prove to be of great service in 



Millionth Measuring Mac/tine. 281 

carrying out the manufacture of an axle, the journal being made 
to a standard gauge, and the bearing being bored out so as to fit a 
difference gauge somewhat larger in size, and of the precise 
difference in diameter, which experience has shown to be neces- 
sary, regard being had to the conditions under which the axle is 
to work. 

It may therefore be conceded that every manufacturer should 
be in a condition to produce difference gauges for use in his 
workshop. 

ART. 219. Sir J. Whitworth has constructed a measuring ma- 
chine with rectangular plane bars in place of the sliding cylinders, 
and with a higher mechanical multiplier, whereby he has measured 
U P tOTTToUoo of an inch. 

We have not space to describe the apparatus fully, but may 
state that the sliding bars are rectangular, with faces made truly 
plane, and that they move in right-angled V-g rooves > whose sur- 
faces are also true planes. 

The ends of the measuring bars are circular planes, each 
about njths of an inch in diameter, and the utmost care is taken 
to ensure that these plane ends shall be truly perpendicular to the 
respective axes of the bars. We have described the method of 
securing this result. 

Of the two measuring bars one is advanced by a screw having 
twenty threads to the inch, and terminating in a graduated hand- 
wheel with 250 divisions on its rim. This gives a quick motion 
of yffVfith of an inch for each graduation. 

The other measuring bar is actuated by a combination of 
screw-gearing. There is first a screw with twenty threads to the 
inch, which terminates in a worm-wheel having 200 teeth upon 
its rim. Then the worm wheel is itself driven by a tangent screw 
carrying a hand-wheel with a micrometer graduation of 250 
divisions upon its circumference. 

It follows that the rotation of the tangent screw through one 
division will advance the bed screw of twenty threads by a space 
equal to 

_ x -- x of an inch, 

250 200 20 

or TOOT, oTToth of an inch. 



282 Elements of Mechanism. 

Results of this character could be extended as far as we pleased 
in theory, but not in practice. The accuracy and truth of the 
pieces upon which we rely are so severely tested that the power of 
human execution soon fails, and hence we can appreciate the 
interest which this apparatus has awakened. 

From the dimensions of the wheels in this machine it has 
been found that a motion of -oooooi inch in the measuring plane 
is equivalent to a motion of '04 inch on the rim of the graduated 
hand-wheel, whence it follows that the machine magnifies the 
motion 40,000 times, or that the eye observes a graduation to 
traverse over a space 40,000 times as great as that which is being 
measured. 

In order to estimate the degree of tightness between the plane 
face of a measuring bar and the corresponding plane surface of 
an object under measurement, a so-called feeling piece or gravity 
piece has been employed. The gravity piece consists of a small 
plate of steel with parallel plane sides, and having slender arms, 
one for its partial support, and the other for resting on the finger 
of the observer. 

One arm of the piece rests on a part of the bed of the machine 
and the other arm is tilted up by the fore-finger of the operator. 
The plane surfaces are then brought together, one on each side 
of the feeling piece, until the pressure of contact is sufficient to 
hold it supported just as it remained when one end rested on the 
finger. This degree of tightness is perfectly definite, and depends 
on the weight of the gravity piece but not on the estimation of the 
observer. 

In this way the expansion due to heat when a 36-inch bar 
has been touched for an instant with the finger-nail may be de- 
tected. 

Also the movement of 'ooooor inch has been indicated by the 
gravity piece becoming suspended instead of falling, and the piece 
has fallen again on reversing the tangent screw through two 
graduations, representing -000002 ; showing the almost infinite- 
simal amount of play in the bearings of the screws. 

For the coarser measuring machine the workman relies upon 
the sense of touch for feeling the size of the body which is being 
passed between the measuring planes. 



Standards of Length. 283 

ART. 220. The national standard of length is a rectangular 
bronze bar, 38 inches long, and i square inch in transverse section. 
A cylindrical hole, f inch in diameter, is sunk near each end to 
the depth of \ an inch ; a second small hole is then bored at the 
bottom of the larger one for the reception of a gold plug, forming 
a table T V inch in diameter, on which three fine lines are engraved 
at intervals of r ^th of an inch in directions transverse to the 
length of the bar. The distance between the two middle lines is 
the length of the standard, and the object of excavating the holes 
is to obtain a measurement along the axial line of the bar. 

This is an example of what is termed line measure, and line 
measure bars are compared by the aid of fixed microscopes, with 
optical contrivances for reading to s^^th f an mcn 

In the year 1834 Sir J. Whitworth obtained two standard yards 
in the form of line measure bars, and by the aid of microscopes 
transferred the mean distance between each pair of engraved lines 
to a rectangular end measure bar, as nearly as he could accomplish 
the task. At the same time he constructed a millionth measuring 
machine for the reception of the bar. 

It now became comparatively easy to subdivide the yard into feet, 
and for this purpose three bars were prepared, each a little longer 
than one foot. A temporary abutment was then raised in the bed 
of the measuring machine, and the bars were reduced and tested 
until (i) they became respectively of the same length, and (2) 
they filled up the length of the standard yard when placed end 
to end. 

Further subdivisions of the foot were made, and finally a 
standard inch was arrived at 

Again, standard end measure bars gave birth to standard cylin- 
drical gauges, and thus the mechanical measures adopted in the 
workshop have been originated and have been reproduced by 
the employment of end measure bars and a good measuring 
machine. 



284 



Elements of Mechanism. 




CHAPTER IX. 



MISCELLANEOUS CONTRIVANCES. 



WE propose to examine in our concluding chapter various miscel- 
laneous pieces of mechanism, and certain special contrivances 
which are of frequent occurrence in machinery, and with which a 
student of applied mechanics ought to render himself familiar. 

ART. 221. The invention of counting wheels is due to the 
celebrated Cavendish, who constructed a piece of apparatus for 
registering the number of revolutions of his carriage wheel. . This 
apparatus is deposited in the Museum of George III. at King's 
College. 

There is but one guiding principle in this branch of mechanism, 
however varied may be the details of the separate parts. 




Each wheel of a series, A, B, C, &c., possesses ten pins or 
teeth, and it is contrived that one tooth only of C shall be suffi- 
ciently long to reach those of B ; similarly B is provided witli one 
long tooth which is capable of driving A. 



Counting Wheels. 285 

Thus C goes round ten times while B makes one revolution, 
and so on for the other wheels ; in this way the series is adapted 
for counting units, tens, hundreds, thousands, &c. 

In fig. 289 the arm EF imparts rotation to the first, or ratchet 
wheel, by means of the paul HD ; the number now registered is 
988 ; after two vibrations of the arm the zero of C will reach the 
highest point, the tooth P will drive B through the space of one 
tooth ; and the number registered will be 990 ; after ten more 
vibrations of the arm, P will again advance B, and at the same 
instant Q will move A, and will bring its zero up to the highest 
point : the three wheels will now register ooo, having passed the 
number 999, which is the last they can give us. 

The wheels are retained in position by the rollers R, R, R, 
mounted upon springs. As each roller is forced in between two 
pins, it not only acts as a detent, but also adjusts the wheel in its 
right position. Mr. Babbage employed this contrivance in his 
calculating machine. 

/ ART. 222. A small counting apparatus is attached to every 
gas-meter used in houses, and registers the number of cubic feet 
of gas consumed ; here, however, the step by step motion is not 
employed, the dial plates are fixed, and a separate pointer travels 
round each dial respectively. 

The pointers are placed upon the successive axes of a train of 
wheels, composed of a pinion and wheel upon each axis, the 
number of teeth on any wheel being ten times that upon the 
pinion which drives it. Suppose, for example, that the pointer on 
the plate registering thousands completes a revolution and adds 
ten thousand to the score, its neighbour on the left will have 
moved over one division on the dial registering tens of thousands, 
and thus an inspection of the pointers throughout the series will 
at once indicate the consumption of the gas. 

These index-fingers move alternately in opposite directions, 
being attached to the successive axes of a train of wheels ; the 
figures upon the counting wheels are also placed in the reverse 
order on every alternate wheel. 

As we are only concerned with the counting apparatus, it is 
not necessary to explain the manner in which the flow of gas 
through the meter sets the train of wheels in motion, but we may 



286 " Elements of Mechanism. 

point out that there is no ratchet wheel employed, and that the 
flow of gas keeps up a constant rotation in an endless screw, 
which starts the train and maintains it in action. 

A reliable counting apparatus which will record the exact 
number of impressions made by a printing machine is indispensable 
in some public departments, and it is found that the best result is 
arrived at by combining a ratchet wheel having a few deep well- 
cut teeth with the train of wheels used in the gas-meter. 

The practice is to place upon the axis of the first or ratchet 
wheel carrying the units a pinion of 10 teeth gearing with a wheel 
of 100 teeth, then another pinion of 10 drives a wheel of 100 teeth, 
and so on, as far as we please. The train of wheels cannot fail to 
record the hundreds, thousands, &c., accurately ; the only possi- 
bility of a mistake occurs with the units, but if the paul be car- 
ried well over the teeth of the ratchet, and if the wheel itself be 
driven at each advance a little beyond the point necessary to give 
another unit, if, in other words, the movement should be a little 
over-pronounced, the register will be perfectly exact. 

In order to avoid the objection that the successive wheels turn 
in opposite directions, an idle wheel is interposed between each 
alternate pinion of 10 and its wheel of 100. All the pointers then 
Devolve in the same direction as the hands of a clock. 

ART. 223. Where it is intended to print the figures regis- 
(/tered, as in the numbering of bank-notes, the step by step motion 
is essential, and, further, each wheel must carry the letters upon 
its edge, and not upon the face ; the apparatus employed is the 
same in principle as that of Cavendish, but the construction differs, 
the wheels being placed side by side and close together. 

In order to present a fair idea of the construction of a num- 
bering machine, that is of a machine designed for printing con- 
secutive numbers, we refer in the first instance to a rough model 
belonging to the School of Mines, and shall afterwards give some 
description of a complete apparatus. 

In the model, the wheels are flat cylindrical discs, having the 
numerals, i, 2, 3, 4, 5, 6, 7, 8, 9, o, painted upon their edges. 
On one side of each disc a ratchet wheel with 10 teeth is carved 
out, while on the opposite face of the disc only one nick or cut rn 
is formed. The cut rn is adjacent to the numeral o, and is in- 



Numbering Machine. 



28; 



tended to serve the same purpose as the projecting tooth in the 
previous arrangement. 

The drawing shows the complete ratchet in side elevation, and 
the position of the teeth with reference to the numbers on the 
disc. Also it will be noticed that the first driver DH is a slender 
bar which encounters the teeth of the ratchet on which it works, 
whereas the second driver NL is twice as broad as DH, and en- 
counters both the ratchet on the second numbering wheel, and 
also that slice of the rim of the first disc on which the nick rn is 
situated. 

FIG. 290. 




It is apparent that so long as NL is resting on the rim of the 
first disc at any part except where the cut rn is formed, it will be 
held above the teeth of the second ratchet, and will be inoperative, 
but that \vhen NL falls into rn it can drive the second wheel. 



288 Elements of Mechanism. 

Thus let the unit wheel mark 9, and the second wheel mark o, 
the number read on the first two wheels being 09, meaning 9. At 
the next stroke NL falls both into rn opposite the numeral o, and 
into the second ratchet at the part opposite the numeral i, 
whereby NL advances both wheels by the space of one tooth, and 
the number 10 takes the place of 09. 

In like manner there is a second cut or nick pm adjacent to 
the numeral o on the second wheel, and on the opposite side to 
the complete ratchet. This determines the period when the next 
driving paul MT shall advance the third numbering wheel, and 
thus the series is continued. 

It follows that the consecutive advance of the respective 
wheels may be provided for by the employment of ratchets, 
having alternately ten teeth and one tooth respectively, and 
placed in regular order upon the numbering wheels, as shown in 
the diagram. 

/V^RT. 224. We can now explain with more particularity the 
//construction of a numbering machine, a considerable portion of 
which is set out in the annexed diagrams. 

The mechanism is automatic or self-acting, the operator grasps 
the handle H, and moves it to and fro as far as it will go ; each 
time that he does so the type is inked, the numbering wheels are 
adjusted, and an impression is taken. 

In order to comprehend the operation we may point out that 
the principal working parts are the following : 

1. The handle H, centred at C, and provided with an arm 
carrying the printing wheels. The central axis of these wheels is 
at P, and since P is rigidly attached to the handle and C is also 
a fixed point, it follows that CP is of constant length. 

2. The two 'cranks, CP, BQ, with the connecting rod PQ, 
whereof CP is an imaginary line, but BQ and PQ control the 
inking apparatus. It has already been stated that advantage may 
be taken of the different; positions of PQ in the above combi- 
nation, and an example is here afforded, as will be seen imme- 
diately. 

3. The numbering wheels lie side by side, and the projecting 
portions, marked^,/, are the successive numerals, o, i, 2 . . . 9. 

4. The inking rollers, marked I, I, work in slots in the arm 



Numbering Machine. 289 

PM, and arc pulled towards P by' elastic bands, not shown in the 
drawing. Connected with the inking rollers is the circular table 

FIG. 291. 




RR, upon which a supply of printing ink is spread out. In the 

working of the machine, the rollers, I, I, run upon the table, RR, 

and receive from it a supply of ink ; they then pass on to the 

u 



290 



Elements of Mechanism. 



faces of the printing type, and supply them with sufficient ink for 
an impression. 

5. There is an impression table, EE, on which the operation 
of printing is performed. 

6. There are two ratchet wheels in combination, being an 
ordinary and masking ratchet wheel respectively, which advance 
the numerals according to some required rate of progression. 
For a description of such wheels we refer the student to Art. 138 ; 
they are not inserted in the present diagrams. 

FIG. 292. 




By comparing the annexed sketch with that given previously, 
the operation of the machine will be clearly understood. 

When the handle H is depressed to the full extent, the num- 
bering wheels are brought down to the printing table, EE, and an 
impression is taken. At the same time the inking rollers run 
back upon RR, and take up a supply of ink. 

During the time that the handle is being raised, the ratchet 
wheels do their work, and advance certain numbers as may be 
required. The inking rollers, in their turn, run from the table, 
RR, to the type, and supply the numerals with sufficient ink for 
the next impression, and thus the process goes on with a degree 
of ease and certainty which it is one of the triumphs of mechanical 
art to accomplish. 



Numbering Machine. 291 

ART. 225. As regards the operation of the ratchet wheels, 
it will be remembered that an ordinary masked ratchet suffices 
to suspend the operation of advancing the unit wheel until two 
strokes have been made, and thus it becomes easy to print each 
number in a series, such as 101, 102, 103, &c., twice over. Also 
after the unit ratchet has done its work, some method embodying 
the principle set forth in previous articles will be employed for 
carrying on the motion to the wheels printing tens, hundreds, 
and so on. 

But there is yet something more to be provided for, inasmuch 
as it is often an advantage to print the odd numbers, as 101, 103, 
105, &c., in one column, and the even numbers, as 102, 104, 106, 
&c., in another column. 

Or the same machine may be required to print consecutive 
numbers. 

The arrangement for effecting this double purpose will be 
understood by referring back to Art. 138, and it will be there seen 
that the numbering wheels are carried by an arm in a circular 
sweep from the inking apparatus to the printing table ; in tra- 
velling along they encounter the paul, which is fixed to the frame- 
work, and if the circle should simply graze, as it were, against the 
paul during its travel, one tooth only would be taken up ; whereas 
by setting the paul so that it meets the circle at a point nearer 
its centre, and strikes it more directly, two teeth may be taken 
up, and thus either one or two units may be advanced at each 
impression. 

i //ART. 226. The ninth chapter was devoted to illustrations of 
the importance of truth of surface, and we may now refer to a 
lecture diagram of a complete machine, which is an ordinary 
example of a combination of elementary surfaces, viz., the true 
plane, the screw, and the cylinder. 

The machine is in use at Woolwich for turning out rapidly the 
bosses or naves of wheels, and is also a direct application of the 
copying principle. 

The block of wood, intended for the nave, is supported be- 
tween centres, as in a lathe, and is rotated rapidly by a belt 
passing over a driving pulley E. The adjacent pulley I rides 
loose on the shaft, and is an idle pulley. 



292 



Elements of Mechanism. 



The cutter c is carried by a slide rest, which resembles the ordi- 
nary slide rest of a lathe. There is a true plane surface DD, with 
inclined sides, supporting the saddle/^. The handle H operates a 
screw which traverses this saddle to and fro along the line parallel 
to the line of centres of the lathe. The form of the nave is 
determined by the cam-groove in the plate AB in which a roller r 
runs as shown in the diagram. A slide carrying the cutter c is 
attached to r, and it follows that the traversing of the saddle fe 
along DD will cause the cutter c to advance or recede in exact 
accordance with the outline of the cam-groove. 

FIG. 293. 




When the block is first put in the lathe it is rough and un- 
even, and the cutter c should be advanced slowly and cautiously, 
in order that it may commence by paring off the principal in- 
equalities. The cutter is therefore brought in or out by a screw 
actuated by a hand wheel A, and at the same time is traversed 
longitudinally by the handle H, the definite position of the cutter 
on the saddle being, of course, quite independent of its motion as 
due to the cam-groove. 

In this way the machine does its work rapidly and effectually, 
the cutter runs to and fro along the outline of the block, and 



Escapements of Watches. 293 

removes the material while copying and preserving the outline of 
the guiding curve. The amount removed is also under the control 
of the operator, and is regulated by the hand wheel h. 
AX'ART. 227. We have reserved for the concluding chapter an 
account of the principal conditions which obtain in the construc- 
tion of the escapements of watches, and have to show that the 
principle of the ^ dead beat' is recognised in the three forms which 
are in common use. 

And here we may remark that the pendulum of a clock 
appears as the balance wheel in a watch. 

A wheel, pivoted on very small steel pivots, and working in 
jewelled supports, is attached to a flat spiral steel spring in pocket 
watches, or to a more powerful helical steel spring in marine 
chronometers. This wheel vibrates under the action of the pull 
of the spring just as a pendulum would do under the pull of the 
earth, but under better conditions theoretically, for the force of the 
spring increases with the angle through which the balance wheel 
swings, and in direct proportion to that angle ; the result, there- 
fore, is that the swing of the watch pendulum is always performed 
in very nearly equal times whether the arc of swing be increased 
or diminished. 

We have what is technically called an isochronous pendulum in 
the balance wheel, and this is important, because the time is not 
affected by small changes in the arc of swing. Further, it should 
be noted that the balance wheel swings through an angle which 
is enormous as contrasted with the swing of the pendulum, being 
more than a whole revolution in the case of the chronometer or 
lever watch. 

Consider now the construction of the chronometer escapement, 
which fulfils our conditions with an exactness that may well sur- 
prise us, and which exhibits in its arrangement a marvellous 
amount of mechanical skill and forethought. 

The detent, which corresponds to the anchor pallets, consists 
of four principal parts : 

1. The locking-stone, D, a piece of ruby, upon which the tooth 
of the escape wheel rests. 

2. The discharging spring, Ar, which is a very fine strip of 
hammered gold. 



294 



Elements of Mechanism. 



3. A screw at A to fix Ar to the stem of the piece SD. 

4. The shank of the detent, consisting of a projecting arm 
Dy, the part DA, and the portion at S, which is cut away to form 
a spring which may bend and act as a pivot on which the whole 
detent can be moved a little. 

The small circle on the left hand has a projecting piece which 
keeps the escapement in action, and it is a part of the stem of the 
balance wheel. 

FIG. 294. 




As the balance wheel swings to and fro, this roller also vibrates, 
and when passing downwards it encounters the spring Ar, and 
pushes it aside without any perceptible effort, because the spring 
bends from the distant point A. 

On its return the projection finds the spring to be capable of 
bending, not from the distant point A, but only from the point g 
against which it rests. The roller therefore takes the spring and the 
whole detent with it and raises the locking stone D from the point 
of the escape wheel, the escape wheel at once flies round, and 
before it can be caught upon the next tooth by the return of the 
detent to its normal position, is enabled to give an impulse to the 
balance wheel by striking against the point d in the manner shown 
in fig. 295. 

The whole arrangement can now be studied from the drawings, 
and is complete with the exception of a banking screw which 
supports the detent when coming back to the position of rest. It 
will be seen that the large circle F is fixed to the smaller one, and 
that the projection marked d is quite clear of the escape wheel 
while a tooth is resting against the detent. 

The advance of the escape wheel is so instantaneous that it is 
not seen to move : it appears to tremble a very little, but it comes to 



The Chronometer Escapement. 



295 



rest again so quickly that the eye cannot follow and can scarcely 
detect the motion. It is, of course, made evident by watching the 
spokes of the wheel. 

FIG. 295- 




What, then, has been the action ? In the first swing of the 
balance the only obstacle has been the bending aside of the spring 
Ar, which is no more than bending a light feather. In the second 
swing the pendulum or balance wheel has had to lift the detent : 
this is a momentary and very small action against it, but as quick 
as thought the action is compensated and the balance receives its 
impulse through equal distances on each side of the middle of its 
swing, according to the principle of the dead-beat escapement. 

Here, then, theory and practice are in exact accord. 

It should be noted that the impulse is given at every alternate 
swing of the balance, and not with every swing as in the case of 
the clock pendulum. 

/ ART. 228. The Lever Escapement comes next in order, and 
here we return to the anchor pallets. The escape wheel is locked 



2c)6 Elements of Mediants in, 

by these pallets, and gives its impulse upon their oblique edges 
in the manner described in Art. 55. 

The balance wheel is free during the greater part of the swing 
(hence the name of Detached Lever), and oscillates through a 
considerable angle. The unlocking occupies an angle of about 
3, and the impulse is given through about 9. 

These are just the conditions which prepare us for the principle 
of the dead beat. 

FIG. 296. 







The pallets, mp, qn, are jewels inlaid into the arms, the light 
steel bar DH is the lever movable about C as a centre. An open 
jaw at one end is capable of receiving a ruby pin, P, attached to the 
roller, which is on the axis of the balance wheel and moves with 
it. There are also banking pins, and a small guard pin to prevent 
the lever from falling out of position. 

As the balance vibrates the pin P swings to and fro with it : 
in doing so it enters the open cut at the end of the lever, and 
removes the locking portion of the pallet from the point of the 
escape wheel. Instantly the escape wheel flies forward, and by 
pressing against one oblique edge suddenly pushes on P, and the 
lever is no longer moved by the balance wheel but imparts an 
impulse to it. Very soon, that is after the nine degrees of the 
swing are consumed, the ruby leaves the lever behind, and the 
wheel goes on detached and unchecked in its swing. 

On its return the pin finds the lever where it had left it, carries 



Escapements of WatcJies. 



297 



it forward, unlocks the escape wheel, receives its impulse, leaves 
the lever behind, and the balance is free for the rest of the 
swing. 

The only action against the balance is that of unlocking the 
escape wheel so as to enable it to give the impulse. This is very 
brief in duration as compared with the whole swing, and the 
watchmaker takes care that it shall be as little as possible. The 
impulse is given just at the middle of the vibration, and the con- 
struction follows out the theory very closely. 

ART. 229. Lastly, we may refer to the escapement of the 
-'so-called Geneva Watches, which is Graham's cylinder movement. 

Here the balance is attached to a very thin cylinder centred 
at o, and the point of a tooth rests upon either the outside or the 
inside of this cylinder during a part of the swing. In this respect 
the action corresponds exactly to the friction of the escape tooth 
against the circular part of the pallets in the dead beat. 

As the cylinder vibrates round its centre o, the tooth pn comes 
under the edge at r, and pushes the cylinder onward : this gives an 
impulse. The tooth soon passes r, flies 
into the cylinder, and is stopped by the 
concave surface near s ; the cylinder 
now vibrates in the opposite direction, 
pn escapes, and in doing so gives 
another impulse at s to the cylinder in 
the opposite direction, and thus the 
action goes on. 

The impulse would not be given in 
the middle of the swing, but through 
small arcs equally distant from the 
middle point, and equal in length to 
each other. Hence this combination is 
nearly identical with the dead-beat es- 
capement, although inferior to it in 
this latter particular. 

The manner in which the effects of the expansion and con- 
traction of the material of the pendulum rod and the balance 
wheel, due to changes of temperature, are rendered innocuous, 
forms a separate branch of the subject. 



FIG. 297. 




298 



Elements of Mechanism. 



ART. 230. The fusee is adopted in chronometers, and in most 
English watches, in order to maintain a uniform force upon the 
train of wheels, and to compensate for the decreasing power of 
the spring. 

The spring is enclosed in a cylindrical barrel, and sets the 
wheels in motion by the aid of a cord or chain wound partly 
upon the barrel and partly upon a sort of tapering drum called 
a fusee. 

As the spring uncoils in the barrel, the pull of the cord de- 
creases in intensity ; at the same time, however, the cord unwinds 

FIG. 298. 




itself from the fusee, and continually exerts its strain at a greater 
distance from the axis, that is, with a greater leverage, and with 
more effect. 

The theoretical form of the fusee is a hyperbola, being the 
section of a right cone made by a plane parallel to the axis of 
the cone. 

To prove this statement we must first recognise the law ac- 
cording to which an elastic body under extension or compres- 
sion exerts a force of restitution whereby it tends to recover its 
original form. 

This law was stated by Dr. Hooke as being contained in the 
maxim ut tensio sic w's, by which it is intended to convey that 
when a body is extended or compressed in a degree less than that 
which produces a permanent derangement of form, the force 
necessary to keep it extended or compressed is proportional to 
such extension or compression. 

Take a spiral steel spring balance, for example ; hang upon it 
successive weights of i, (2), (2 + 1), (3 + 1) Ibs., the index point 




The Fusee. 299 

will descend through equal spaces for each additional pound 
weight, and will rise by equal spaces as each pound is successively 
removed. 

Assuming the law to hold exactly when the spiral spring is 
subject to a force of torsion instead of one of direct extension, we 
shall have the pull of the spring proportional to the angle through 
which the barrel has been made to turn. 

Let DPBA represent one-half of the sec- 
tion of a fusee, DPB being the curve whose 
equation is to be found. 

Draw DE, PN, BA perpendiculars on 
EA ; take ER, QN, SA to represent the pull 
of the spring when the chain is at the points 
D, P, and B, respectively. 

According to Hooke's law, the force of the 
spring will decrease uniformly as the chain 
passes from D to B, therefore RQS must be 
a straight line inclined to EA. Produce 
RQS to meet EA in C. 

Then ^rr = ;pr.> which is a constant ratio, by reason of the 
V^.N VX.A. 

law of elasticity. 

Assume that this ratio is represented by m, 
.-. QN = m . CN. 

In order that the fusee may accomplish its object, the product 
of the pull of the spring into the arm NP must remain constant 
for every position of P. 

Hence, calling CN = x, NP = y, we have 
(pull of spring) x NP = ;// . CN x NP = mxy. 

But this product is not to vary, 

/. mxy = a constant quantity, 

or xy = a constant, 
which is the equation to a hyperbola. 

In practice, where great accuracy is required, the strength of 
the spring is tested by fixing a light lever to the winding square 
of the fusee, and observing whether the pull of the spring is 



300 



Elements of MecJianism. 



balanced in every position by the same weight hung at the end of 
the lever. The fusee would be cut away a little where it was 
_4^eSsary to do so. 

jr ART. 231. In mechanism the fusee is frequently employed 
to transmit motion instead of to equalise force, and enables us to 
derive a continually increasing or decreasing circular motion from 
the uniform rotation of a driving shaft. 

The groove of the fusee may be traced upon a cone or other 
tapering surface, or it may be compressed into a flat spiral curve : 
in all cases the effect produced will be that due to a succession 
of arms which radiate in perpendicular directions from a fixed 
axis, and continually increase or decrease in length. 

The fusee can of course only make a limited number of turns 
in one direction. 

A flat spiral fusee occurs in spinning machinery, and serves 
to regulate the velocity of the spindles, and to ensure the due 
winding of the thread in a succession of conical layers upon a 
bobbin or cop. 

The formation of the cop is a problem upon which a vast 
amount of mechanical ingenuity has been expended ; and without 



FIG. 300 FIG. 301. 



entering too much into details, we may ob- 
serve that there are two distinct stages in the 
process of winding the yarn upon a spindle 
so as to produce a finished cop. 

The copbottom (fig. 300) is first formed 
upon a bare spindle by superposing a series 
of conical layers with a continually increasing 
vertical angle. 

The body of the cop is then built up by 
winding the yarn in a series of equal conical 
layers. (Fig. 301.) 

The winding-on of the yarn begins at the 
base of the cone and proceeds upwards to 
the vertex ; the spindles are driven by a 
drum which rotates under the pull of a chain, 
and they may be made to revolve with in- 
creasing rapidity by placing a fusee upon the 
driving shaft and causing the chain to coil upon it. 




The Fusee. 



301 



FIG. 302. 



Such an arrangement as shown in fig. 302 will be adapted to the 
winding of a uniform supply of thread upon a conical surface ; and 
we can easily compre- 
hend that a fusee of fixed 
dimensions will do veiy 
well for building up the 
body of the cop after the 
foundation is made. The 
main difficulty occurs in 
producing the copbot- 
tom, where the series 
of conical layers of con- 
tinually increasing ver- 
tical angle demands a 
fusee whose dimensions 
shall gradually contract towards the centre. 

The method of contracting the form 
explained as follows : 

Fig. 303 represents portions of two flat discs having axes at A 
and B, and upon which are cut radial and curved grooves in the 

FIG. 303. 




of the fusee may be 




manner indicated ; it being arranged that when one plate is placed 
upon the other, the pins P and Q shall travel in both sets of 
grooves at the same time. 

We can easily see that the blocks which carry the pins will 



302 



Elements of Mechanism. 




move along the radial grooves as the disc B turns relatively to A, 
and that by this combination we can obtain a spiral fusee of 
any required form, and can contract or enlarge its dimensions at 
pleasure. 

ART. ^^. If two cords be wound in opposite directions 
round a drum, A, and the ends of the cords 
be fastened to a movable carriage, it is evi- 
dent that the rotation of A in alternate di- 
rections will cause a reciprocating movement 
in the carriage. 

This is a mangle in its simplest form, and 
the objection that the handle must be con- 
tinually turned in opposite directions may be obviated by the use 
of the mangle wheel. 

It is clear that if the drum were divided into two portions, and 
that each half instead of being cylindrical were formed into a 
fusee, the motion of the piece driven by the rope would be no 
longer uniform but would vary with the dimensions of the fusee. 
Hence the drum A has been replaced by a spiral fusee in the 
self-acting mule of Mr. Ro- 
berts, and thus the motion of 
the carriage is gradually acce- 
lerated until it has reached the 
middle of its path, and then 
decreases to the end of the 
movement. 

It must be understood that 
the cord fastened at A goes off at C, while that fastened at B 
passes on to D. 

FIG. 306. 





A helical screw of a varying pitch traced upon a cylinder would 
produce a similar variable motion of the mule-carriage, and has 



Winding-on Motion. 



303 



FIG. 307. 



been applied in a machine constructed upon a different principle, 
in order to obtain a continually decreasing motion of the carriage. 
It replaces the fusee. 

ART. 233. Mr. Robertas winding-on motion reposes upon the 
principle of the fusee, though in a modified form. 

Let one end of a rope which is coiled round a drum be at- 
tached to a point, P, in the movable arm CP ; it is evident that 
the rotation of CP about the centre of motion C will cause some 
portion of the rope to be unwound from the barrel (fig 307). 

Draw CS perpendicular to 7 the direction of the rope ; then, at 
any instant of the motion/' / 
the arrangement supplied ^^ 
the jointed rods, CP, BQ, \"<\ 
mentioned in Art. 93, and 
it is manifest that the rate 
at which the string is ^- j 
wound will vary as the per- 
pendicular CS. *\ 

This ratek is greater t 
when CP\is perpendicular 
to PQ\ bu\ decreases to 
nothingVvhen x GS vanishes, 

and here\ therefore, the varying arm of the fusee exists in a latent 
form. \\ 

Next conceive that the conditions are changed, and that the 
drum B moves to the right hand through a moderate space, while 
CP remains fixed. The cord will unwind from the drum with a 
nearly uniform velocity. 

If, finally, the arm CP be not fixed, but be made to move from 
a position a little to the left of the vertical into one nearly hori- 
zontal during one journey of the drum, it is abundantly clear that 
we shall subtract from the uniform motion of unwinding that 
amount which is due to the action of a fusee, and that if the 
spindles derive their motion from the rotation of the drum they 
will continually accelerate as the drum recedes from CP. In this 
way we can make up the " body of the cop. To form the cop- 
bottom, it is necessary that the winding on should begin more 
rapidly, and should gradually diminish. This character of motion 




304 Elements of Mechanism. 

is produced by causing the nut P to traverse CD in successive 
steps during each journey of the drum. As soon as the cop has 

FIG. 308. 




attained its full diameter the nut ceases to travel along CD, and 
the thread is wound in uniform conical layers. 

ART. 2^.. Harrison's going fusee is employed in watches or 
clocks having a fusee for the purpose of keeping the timepiece 
going while the spring is being wound up. 

The principle on which it is constructed will be readily under- 
stood. Referring to the small sketch in fig. 309, conceive that a 
force applied in direction of the arrow at x is to be communicated 
to y. Let x and y be connected by a spring S, and suppose 
further that some resistance to motion is felt at y, then the force 
at x would compress the spring until y began to yield, after which 
x and y would move together as if they were one piece. 

Next suppose that x becomes fixed, then it is apparent that 
the elasticity of the compressed spring S will cause a push to be 
felt at j, and that for a short interval the spring would urge y 
onward just as if the driving pressure on x had been maintained 
unimpaired. In other words, if we interpose a spring between the 
driver and the thing driven, it is possible to obtain some amount 
of working pressure from the spring after the driving force has 
ceased to act. 

We pass on to explain the drawing, which is taken from a 
model arranged for making the contrivance intelligible to a 
learner. If the construction adopted in a watch were more closely 
followed it would be difficult to show the working parts. 

A weight W, hanging upon a string wound round the disc A, 



The Going Fusee. 



305 



supplies the pull of the chain on the fusee, and the disc is pro- 
vided with a ratchet, called the winding ratchet, having an ordi- 
nary detent at R. On the same axis as A is a circular plate B, 
having a ratchet, called the going ratchet, which is prevented 
from recoiling by a detent P. Connected with A, and forming 
part of it, is the great wheel D, which is shown in the model as a 
pitch circle without teeth, and which drives the pinion E. 

FIG. 309. 





The object is to keep E rotating in the direction of the arrow, 
and from what has been stated it is apparent that so long as W 
acts upon A it will act also on D and will drive E. The difficulty 
occurs when the weight is being wound up, in which case A is 
turned in the reverse direction, and D would be powerless to drive 
E, unless some new agent were called into play. 

In the model this agent is supplied by the strong indiarubber 
cord ST, and in a watch it takes the form of a curved strip of 
steel, whose ends are brought near together. A circular slot 
marked by a dark thick line terminating in T determines the 
amount through which the recoil of the spring can act, and it 
should be noted that ST is attached at S to a pin on the plate B, 
and at T to a pin in the plate D. 

The first action of W is to stretch ST, and when the spring is 
sufficiently stretched D begins to move and travels round, the 
going ratchet slipping tooth by tooth under the detent P. When 



306 



Elements of Mechanism. 



the winding-up takes place and the pressure of W is taken off, the 
action of the spring begins, and there is a pull of S towards T, 
and of T towards S. The detent P, acting on the going ratchet, 
prevents any motion in the direction ST, and therefore T ap- 
proaches towards S, but in doing so ST pulls D in the same 
direction as that in which W previously moved it, and the result 
is that the motion of E continues, notwithstanding that the weight 
W has ceased to act. 

ART. a|2> The mechanism of a keyless watch, so far as the 
winding of the spring and the setting of the hands are concerned, 
may be of interest to the student. 

There are three separate things to be provided for : 

1. The spring is to be wound up without opening the case or 
inserting a key. 

2. The same button or handle which turns the spring should 
be available for setting the hands. 

3. No injury should happen if the winding button be turned 
in the wrong direction. 



FIG. 310. 




Taking number 3 first, the contrivance adopted is a very old 



Keyless Watch. 307 

one in mechanism, being a ratchet cylindrical coupling, whereof 
the two parts are held together by a spring. 

It is shown at R, in figs. 310 and 311, the spring being marked 
Q, and terminating in a tail, which holds the coupling by the 
groove S. When the spring is in action the two halves of the 
coupling are locked together, but upon depressing the stud P the 
tail S is forced down, and the halves of the coupling are separated. 
The button H, which is rotated for winding the watch-spring, or 
for setting the. hands, is connected directly with RC, the lower 
half of the coupling, and when the spring is in action the turning 
of RC causes L also to rotate. 

Thus, when the button is rotated in the direction of the arrow, 
the lower half of the coupling drives the upper half, but when the 
button is rotated in the opposite direction, the ratchet teeth on 
the lower half slip over those in the upper half, and all that 
happens is that RC vibrates up and down by the depth of a tooth. 
The movement is harmless, for no part of the mechanism which 
does any work is affected by it. 

The winding square in an ordinary watch is replaced by the 
spur wheel D, having a detent r, actuated by a spring, whereby 
the toothed wheel serves the same purpose as a ratchet wheel with 
saw-cut teeth. This contrivance is borrowed from the engineers, 
as will be remembered. 

The spur wheel D gears with another spur wheel B, on the face 
of which is a flat bevel wheel which engages with another bevel 
wheel L attached to the upper half of the ratchet coupling. 
The rotation of H in the direction of the arrow drives RC, which 
again drives L, and so causes B and D to rotate, and thereby 
to wind up the watch. 

The setting of the hands is accomplished by a wheel A which 
operates on the minute hand. The lower half of the coupling 
RC terminates in a cylindrical toothed wheel, technically known 
as a crown or contrate wheel, which is brought into gear with the 
spur wheel A when the stud P is depressed. It will be apparent 
that when RC is lowered the winding-up stops, and at that time 
the hands can be set. Whereas when RC is allowed to rise the 
crown wheel is thrown out of gear with A, and the winding-up 
Can begin. Each action is shown separately in the sketches. 
x 2 




308 Elements of Mechanism- 

ART. 236. The snail is chiefly found in the striking part of 
repeating clocks. It is a species of fusee, and is used to define 
Flo I2 the amount of angular deviation of 

a bent lever ABD, furnished at the 
end A with a pin which is pressed 
against thk - curve of the snail by a 
spring, and\is attached at the other 
end to a curv&d rack, whose position 
determines the number of blows 
which will be struck upon the bell. 

In order to form the snail, a 
circle is- divided into a number of 
equal parts (twelve, for ^exiample), and a series of steps are formed 
by cutting away the plate and leaving a circular boundary in each 
position. 

As the snail revolves, ABf) jpasses by jerks into twelve different 
positions, and the clocklstrikes the successive hours. 

Since the point A Describes a circle about B, it is clear that 
the depth of each step must vary in order to obtain a constant 
amount of angular motioii in the arm B A during each progressive 
movement. It will be seehjhat the circular arc described by the 
end of the small lever has its tangent at A, when in the position 
sketched, parallel to the vertical diameter which divides the snail 
into two equal parts, and this reduces the inequality between the 
steps. 

ART. 2. The disc and roller is equivalent to the fusee, and 
is now but nttle used, on account of the probability that the roller 
will occasionally slip. 

This arrangement consists of a disc A, revolving round an axis 
perpendicular to its plane, and giving motion to a rolling plate B, 
fixed upon an axis which intersects the axis of the disc A at right 
angles. 

Supposing the rotation of the disc to be uniform, that of the 
roller B will continually decrease as it is shifted towards the centre 
of A, and conversely. 

This is precisely the effect produced by a fusee. 
The roller may be a wheel furnished with teeth, and may roll 
upon a spiral rack, as shown in the diagram. 



The Disc and Roller. 



309 



As the disc revolves the pinion P slides upon the square shaft, 
and is kept upon the rack by the action of a guide-roller, R, which 
travels along the spiral shaded groove. 

FIG. 3I3 . 





This example is by no means put forward as a good mechanical 
contrivance, for indeed the disc and roller possesses an inherent 
defect which should be diminished as much as possible in practice 
and not exaggerated. The bounding circles of the roller run with 
the same linear velocity, whereas the circular paths upon which 
they are both respectively supposed to roll move with different 
linear velocities, by reason of their being concentric circles of un- 
equal size traced upon a plate which rotates with a uniform angular 
velocity about an axis through the common centre. 

It is geometrically impossible that the bounding circles of the 
roller can both roll together, or the combination will fail in exact- 
ness as a piece of pure mechanism, and thus two rollers running 
round upon a flat surface or bed-stone in the manner suggested 
form an excellent pulverising or grinding apparatus. These rollers 
are called edge-runners : they are of large size and very weighty, 
and are placed near to the vertical axis about which they run. 

It is apparent, without any proof, that the disc B will change 
the direction of its rotation as soon as it has passed over the 
centre of the plate A. This follows from the nature of circular 
motion, and, as we have indicated, the disc B will come to a 
standstill when its circumference reaches the centre of the driving 
plate A. 

In like manner, if the axis of B be carried along in a straight 



3 1 o Elemen is of Median ism. 

line passing over the centre of A, the student will understand thai 
if A rotates in one direction while B is approaching its centre, anc 
in the opposite direction, as soon as B has crossed the centre, ii 
necessarily follows that the reciprocation of A will impart a con- 
tinuous rotation in one direction to the roller B. 

FIG. 314. 




Such a result is exhibited in the sketch on the left-hand side 
of fig. 314, where the arrow shows that A is rotating in one direc- 
tion when B is above the centre C, and in the opposite direction 
when B is below the same centre ; while it becomes evident that 
the direction of rotation of B is correctly indicated by the arrow 
at P, and remains invariable. 

ART. 238. This movement has been applied in the construc- 
tion of a continuous indicator. It has been shown that Watts 
indicator gives the amount of work done in any stroke of a steam 
engine, and it follows that the general performance of the engine 
would be inferred from a comparison of several indicator cards 
taken at intervals. 

The continuous indicator by Messrs. Ashton and Storey will, 
however, furnish a complete register of the work done in a_given 
time. In this apparatus the piston of the indicator is attached tc 
the roller B, and the plate A takes up, on a din)rmshed scale, the 
reciprocating motion of the piston of the engine. 



Continuous Indicator. 311 

A general idea of the mechanism of the instrument may be 
gathered from the diagram. A grooved wheel, M, is connected 
directly with the disc A, and receives a reciprocating motion from 
the piston of the engine just as if it were the barrel of an ordinary 
indicator. The roller B is connected on the side marked P with 
the indicator piston, and travels up and down in the vertical line 
PE, the motion of B being subject to the opposing forces of a' 
steel spring and the steam pressure, just as if it were the pencil in 
an ordinary indicator. The axis EP is provided with an elongated 
pinion, as shown, which gears into a wheel H in every position, 
the intent being that the rotation of B shall be carried by H to a 
recording or counting apparatus at the top of the instrument, and 
in the line ba produced. 

The principle relied on is that the work done by the engine 
in a given time will be directly proportioned to the number of re- 
volutions made by H in the same time. 

Taking a condensing engine as an example, it is apparent that 
when trW indicator piston is in the position which accords with 
the tra/ing of ~an atmospheric line in an ordinary diagram, the 
disc BHies exactly over the centre of the plate A and does not 
rotate. In subh a state of things the wheel H also remains at rest 
and no work is done. Whereas any increase of the steam pres- 
sure carries B above the centre of A, and causes H to rotate with 
a velocity which increases as the pressure rises. Or, if the vacuum 
be improved, B sinks to a greater distance below the centre of A, 
and H rotates more rapidly during the return stroke, the result 
being t register work done in both cases. Also, if at any time 
there-should be a subtraction of area in an ordinary indicator 
diagram, there will be a like subtraction here by reason that B 
wlrV- cross over the centre before the direction of rotation of A 
is changed, whereupon H will begin to rotate in the reverse 
direction, and some of the work previously scored up will be, as 
it were, rubbed out. Hence, the instrument sums up, or inte- 
grates, as it is termed, the actual performance of the engine in any 
given interval of time. 

ART. fi^ Step wheels constitute a modification of toothed 
wheels ; they are due to Dr. Hooke, and are used to ensure a 
smooth action in certain combinations of wheel-work. 



312 Elements of Mechanism. 

It is evident that the action of two wheels upon each other 
becomes more even and perfect when the number of teeth is in- 
creased, but that the teeth at the same time become weaker and 
less able to transmit great force. 

Dr. Hooke's invention overcomes the difficulty, and virtually 
increases the number of teeth without diminishing their strength. 

Several plates or wheels are laid upon one another so as to 
form one wheel, and the teeth of each succeeding plate are set a 

FIG. 315. little on one side of the preceding one, it being 
provided that the last tooth of one group shall cor- 
respond within one step to the first tooth of the 
next group. The principal part of the action of 
two teeth occurs just as they pass the line of centres, 
and there are now three steps instead of one from 
the tooth A to the tooth B. 

A single oblique line might replace the succes- 
sion of steps, but we should then introduce a very objectionable 
endlong pressure upon the bearings of the wheels. 

Pinions of this construction are to be met with in planing ma- 
chines, and are employed to drive the rack which is underneath 
the table ; so again step wheels are used in marine steam engines 
where the screw shaft is driven from an axis considerably above it. 
They are valuable where strength and smoothness of action are 
to be combined. 

ART. 240. It has been shown that in the case of ordinary 
bevel wheels the pitch cones have a common vertex, and there- 
fore of necessity the axes of rotation meet in a point. In some 
cases, however, it may be convenient that the axes of bevel wheels 
should pass close to each other without intersecting ; the teeth 
have then a twisted form, and the wheels are known as skew 
bevels. 

A general idea of the principle of construction adopted in 
wheels of this kind may be obtained by a simple experiment. 
Fasten a light rod of wood or a bright steel wire, such as a knitting 
pin, obliquely to the face-plate of a lathe, in such a manner that 
the direction of the rod intersects at an angle the axis or line of 
the lathe. Set the plate in rapid rotation, when the rod will 
appear to be replaced by a shadowy double cone. The effect is 



Skciv Bevels. 



313 



due to the fact that any impression made on the retina of the eye 
endures for an appreciable time. 

Now fix the rod in such a manner that it no longer cuts the 
line of centres, but passes obliquely on one side thereof, and set 
the face-plate in rapid rotation as before. The double cone will 
be replaced by a surface resembling a dicebox, which is curved 
in every part, and yet is generated by a straight line. This sur- 
face is known to mathematicians as a 'hyperboloid of revolution.' 
From the mode of generation it is clear that the surface in question 
is symmetrically disposed about the line of centres of the lathe, 
and we may regard that line as its axis. 

If two such hyperboloids be brought into contact along a pair 
of generating lines they will roll together, and will serve to com- 
municate a motion of rotation from the axis of one to that of 
the other ; that is, they will enable us to communicate motion 
between two axes which are not parallel and do not meet, the 
directions of the generating lines in each surface determining the 
general direction of inclined teeth which might lie on the re- 
spective surfaces. Thus two hyperboloid surfaces replace, in the 
case of skew bevels, the pitch cones of ordinary bevelled 
wheels. 

ART. 241. Root's Blower, a mechanical equivalent for a fan, 
is a special contrivance which has been modified and varied by 
the ingenuity of patentees, and has been extensively used. 



FIG. 316. 



FIG. 317. 




The sketch fig. 316 is from a model in the collection of the 



3 1 4 Elements of Mechanism. 

School of Mines, and shows the principle of the action of this 
form of air-compressing machine. 

There are two rotating pistons, B and D, centred on axes at 
C and F, and driven by a pair of equal spur wheels in opposite 
directions with equal velocities. A circular case surrounds the 
pistons, and it is provided that there shall be no actual contact 
of the surfaces of the rotating pistons, either with each other 
or with the casing, but that the working parts shall approach as 
closely as may be without any rubbing action or internal friction. 

In the diagram the arrows show the manner in which the air 
enters the casing at A, and is swept forward to the exit at E, the 
two small arrows on the pistons being employed to indicate their 
relative directions of rotation. 

In fig. 317 the blower is converted into a ventilating appa- 
ratus for a colliery, and is in use at a pit belonging to the South 
Durham Coal Company ; it consists of two pistons, each 25 feet 
in diameter and 13 feet wide, driven by engines having a pair of 
28-inch cylinders with 4-feet stroke. 

The clearance between the periphery of one of the pistons 
and the central circle of the other is inch, and it is stated that 
when the ventilator runs at 21 revolutions per minute the amount 
of air passing through the machine is about 118,000 cubic feet per 
minute. 

ART. 242. The Governor of a steam engine usually appears 
under the form invented by Watt, and has proved of the greatest 
possible value in steam machinery. 

The diagram shows this well-known piece of apparatus, 
and the principle of its action may be described very briefly as 
follows : 

The engine imparts rotation to the balls of a heavy conical 
pendulum, and maintains them at a certain inclination to the 
vertical ; if the velocity of the engine be increased, the balls open 
out more widely ; if it be diminished they collapse, and in doing 
so they set in motion a system of levers which is connected with 
a throttle valve, and thereby regulate the supply of steam to the 
cylinder 

i. A common method of constructing the governor has been 
that shown in fig. 318. The balls are suspended at the points E 



The Governor of a Steam Engine. 



315 




TO VALVE 



and H, a little on either side of the central vertical spindle CB. 

Each arm, as HD, is connected by 

a link to a sliding block ST. As 

the rate of rotation increases the 

balls fly out, ST rises, and in doing 

so actuates a lever which controls 

a steam valve and diminishes the 

supply of steam. 

The arm DH is produced to 
meet the vertical axis in C, and DB 
is drawn perpendicularly to CB, 
whence the balls and suspending 
arms lie upon the surface of a cone 
whose axis is CB. The chief point 
to notice is that the number of re- 
volutions made per minute by the 
balls depends upon the height of the cone, viz., CB. 

The effect of placing E and H at a little distance from the 
axis CB is to cause the variation in the height of the cone to 
become greater for any given rise of the balls, and thereby to 
render the governor less sensitive. Thus the heights of the cone 
in the two positions shown are CB and cb respectively, the vari- 
ation being equal to Cc + B& 

Special methods of construction have been originated for 
reducing the amount of this variation ; as to which the reader is 
referred to the author's text book on the Steam Engine. 

An engineer can easily arrange that the variation in speed 
admitted by the governor shall not exceed one-tenth of the mean 
velocity, but it is of the essence of the invention that some change 
in the speed should be admissible : the balls cannot alter their 
position unless the time of a revolution changes, and they cannot 
accumulate such additional momentum as may be sufficient to 
move the valve until the rate of the engine has sensibly altered. 

In some cases, as where the engine drives machinery for very 
fine spinning, it may be desirable to obtain an almost absolute 
uniformity of motion ; or, again, it may be an object to avoid the 
fluctuations in speed to which the common governor is liable 
when any sudden change occurs in the load upon the engine. 



316 



Elements of Mechanism. 



ART. 243. In order to control the engine with almost theo- 
retical exactness, and to provide against the objections to which 
Watt's governor is exposed in certain extreme cases, Mr. Siemens 
has put forward a remarkable adaptation of epicyclic trains to the 
conical pendulum ; and we shall proceed to an examination of his 
invention. 

The original construction of this governor is exhibited in the 
diagram, and is better adapted for the purposes of explanation than 
a more recent arrangement (fig. 319). 

An epicyclic train of three equal wheels, A, B, C, is placed 
between the driving power and the conical pendulum ; of these, 
A is driven by the engine, C is connected with the pendulum, and 
B is capable of running round A and C to a small extent defined 
by stops, the joints at F and B being so constructed as to permit 
of such a motion. 

FIG. 319. 

JW>^ 

A 





The wheel, B, is also con- 
nected by the system of levers 
to a weight, K, and shuts the 
steam valve when its motion has 
lifted K through a certain space. 

The valve spindle passes 
through the centre of motion, E, 
and is turned by the arm FE. 

A conical pendulum, DP, is 
suspended by a ball-and-socket 
joint at S, and the extremity D 
moves in a circular groove, DH. 



T/ie Chronometric Governor. 317 

In this way the rotation of C is communicated directly to the 
pendulum. 

It will be seen that a certain amount of energy is absorbed in 
preserving the pendulum at a constant angle with the vertical, and 
it is a part of the contrivance to increase artificially the friction 
which opposes the motion of the pendulum, and thus finally to 
make the pressure exerted by the weight, K, an actual measure of 
the amount of the maintaining force. 

The governor is at work when the velocity of the engine is just 
sufficient to keep K raised through a small space. 

In order to understand the peculiar action introduced by the 
epicyclic train we should remember that one of these two things 
will happen : either A and C will turn at the same rate, or else B 
will shift its position and run round the axis AH ; there can be no 
departure from the rigid exactness of this statement. 

Now, the wheel C is connected with the pendulum, and its 
rotation cannot be maintained without a constant expenditure of 
energy ; in other words, the tendency of C is to lag behind A, 
and to cause B to run round the axis AH. 

This indisposition in C to accept the full velocity of A is 
artificially increased by the friction until B shifts its position and 
raises the weight K permanently, and then of course it follows that 
the pull of K evidences itself as a constant pressure tending to 
drive the wheel C. 

The pendulum being in this manner retained in permanent 
rotation, suppose that any increase were to occur in the velocity 
of A : the wheel C is in connection with a heavy revolving body, 
and can only change its velocity gradually, but K is already lifted, 
in the sense of being counterpoised, and the smallest increase of 
lifting power can therefore raise it higher ; thus the tendency to 
an increase in the velocity of A will at once cause B to change its 
position, and will control the steam valve. 

So sensitive is this form of governor to fluctuations in speed, 
that an alteration of ^th of a revolution may suffice to close the 
throttle valve altogether. It is in its power to move the valve, as 
well as in its sensitiveness, that this arrangement presents so re- 
markable a contrast to Watt's governor, where the moving force 
on the valve spindle is only the difference between the momenta 



3 1 8 Elements of Mechanism. 

stored up in the two positions of the balls. In this form of 
governor the power is only limited by the strength of the rods and 
levers ; for it is apparent that the whole momentum stored up in 
the revolving pendulum would in an instant be brought to bear 
upon the valve spindle if any sudden alteration were to occur in 
the velocity of the wheel A. 

In the method which has been adopted at Greenwich for 
registering the times of transits of the stars by completing a gal- 
vanic circuit at the instant of observation, a drum carrying a sheet 
of paper is made to revolve once in two minutes. A pricker 
actuated by an electro-magnet, and moving slowly in a lateral 
direction, is set in motion at the end of each beat of the seconds 
pendulum of a clock, and thereby makes a succession of punctures 
in a spiral thread running round the drum. The observer touches 
a spring at the estimated instant of the time of transit of a star 
across a wire of the telescope, and, producing a puncture inter- 
mediate to those caused by the pendulum, does in fact record the 
exact period of the observation. The regularity of motion in the 
drum is a matter of vital importance, and was at one time ensured 
by the employment of a clock train moving under the control of 
this pendulum of Mr. Siemens. 

But, inasmuch as the tendency to simplify mechanical move- 
ments is always leading to new results, it has been found advan- 
tageous to replace the more complicated governor, with its train of 
wheels, by a conical pendulum controlled by a so-called dipper. 
The pendulum is driven by an ordinary clock train, and the 
dipper is carried round with the ball, and is merely a small flat 
plate dipping partially into a bath of glycerine and water. When 
the pendulum is accelerated the dipper enters the liquid more 
deeply, and the drag is greater, whereas when the velocity of the 
pendulum diminishes the dipper rises, and the motion is less 
retarded. It is said that this apparatus fulfils its purpose ex- 
tremely well. 

ART. 244. The Double Eccentric for Reversing an Engine. 
When the piston is near the middle of its stroke in a locomotive 
or marine engine, the slide-valve will have moved over the steam- 
ports in the manner pointed out in fig. 320. 

The slide-valve is connected with a point in the circumfer- 



TJic Double Eccentric. 



319 



ence of a small circle which represents the path of the centre 
of the eccentric pulley, and the piston is connected with a point 
in a larger circle, representing the path of the centre of the 
crank-pin. 

The piston and valve are shown as separated in the drawing, 
but the small circle is repeated in the position which it actually 
occupies, and the method of reversal is the following : 

In the upper diagram the piston is supposed to be moving to 
the right, and the valve to the left, the piston having travelled so 
far in its stroke that the valve is returning to cut off the steam : 
in order, therefore, to change the motion, we must drive the piston 
back by admitting the steam upon the opposite side, and by 
letting out that portion of the steam which is urging the piston 
forward. Hence we must move the valve into the position 
shown in the lower diagram, and shift the centre of the eccentric 







pulley from A to B : the piston will then return before it reaches 
the end of the cylinder, and the movement of the engine will 
be reversed. 

In examining the diagram it should be understood that the 
crank which works the slide-rod is inclined at an angle somewhat 
greater than 90 to the crank which is attached to the piston ; and 
also that the crank of the slide-rod is always in advance of the 
larger one in its journey round. The engine would not work if 



320 Elements of Mechanism. 

the larger crank were to turn in the opposite way to that shown in 
the sketch. 

This explanation shows that in reversing an engine we must 

either shift the eccentric from the one position into the other, or 

else we must employ two eccentrics, and provide some means of 

connecting each of them in turn with the slide-valve. 

V ART. 245. The link motion commonly appears under three 

(\ "forms, (i) There is the shifting link, having its concave side 

* towards the axle or crank shaft : this arrangement was introduced 

by the celebrated Stephenson, and is known as Stephenson's Link 

Motion. (2) There is the stationary link, where the curvature is in 

the opposite direction. (3) There is the straight link, which is 

derived from a combination or moulding together of the two 

former contrivances. 

i. Stephenson's link motion is shown in fig. 321. 

AB is the starting lever, under the control of the engine-driver, 
and is represented as being pushed forward in the direction in 
which the engine is moving ; CD is the link, provided with a 
groove, along which a pin can travel ; a short lever, centred at R, 
is connected at one end, Q, with the slide-valve, and at the other 
end with the pin which moves in the link. 

It is clear that so long as the pin remains near the point D, 
the lever centred at R will be caused to oscillate just as if the pin 
were attached to the extremity of the outer eccentric bar, and that 
the outer eccentric alone will be concerned in the motion of the 
valve. 

If now the engine-driver wishes to reverse his engine, he pulls 
back the lever AB, and by doing so he raises the link CD until 
the pin comes opposite to the end of the inner eccentric bar. The 
raising of the link is caused by the motion imparted to the bell- 
crank lever, GEF, which is centred at the point E. A counter- 
poise to the weight of the link is attached to the axis passing 
through E at some little distance behind the bell-crank FEG, so 
as to be out of the way of the moving parts, and the object of this 
counterpoise is to enable the engine-driver to raise the heavy link 
and bars easily. 

The inner eccentric bar now alone comes into play, and the 
two eccentrics being fastened to the crank axle at the angles 



Reversing an Engine. 



321 




322 Elements of MecJianisin. 

indicated in the first part of the article, it is apparent that the 
valve will be shifted, and that the action of the engine will be 
reversed. 

1fhe stationary link shown in fig. 322 was invented by Mr. 
:h, of the Great Western Railway. It will be seen that the 
'link RS is here suspended by an arm 1>C, so as to be stationary so 
far as any up-and-down movement is concerned, and that it is 
circular in form, being struck by a radius equal to DR, whereby 
also its concavity lies towards the cylinder and away from the axle 
or shaft of the engine. In the sketch the forward eccentric is in 
operation, and the motion is readily traced from the axle to the 
slide, which is shown as having partly uncovered the steam port 
marked A. On pulling the rod H, which is in connection with 
the starting lever, or its equivalent, the bell crank K<?L is moved, 
and the jointed rod DR is brought down by the pull of LM into 
a lower position, whereby it imparts to the slide the motion due 
to the back eccentric, and the engine is consequently reversed, 

3. A third method is Allan's straight link motion, in which the 
link and the valve rod are both shifted in opposite directions at 
the same time. When the link is shifted it must of necessity be 
curved towards the eccentric rods, and when the slide rod is 
jointed as at D, and shifted up or down, the curvature of the link 
must be towards the slide, from which it follows that if both the 
link and the slide rod shift in a vertical plane the concavity and 
convexity may neutralise each other, and a straight link may serve 
to give the motion. Link motions prove to be rather complicated 
pieces of mechanism when any attempt is made to analyse them 
thoroughly, and therefore it may suffice to say that with a sta- 
tionary link the lead of the slide is maintained constant under all 
changes in the position of the sliding block, whereas with the 
shifting link the lead increases a little towards the central position. 

A special advantage of the several link motions consists in the 
power which it gives to the engineer of regulating the supply of 
steam admitted into the cylinder. By moving the starting lever 
or its equivalent into intermediate positions, the amount of travel 
of the valve is reduced at pleasure, for it is evident that no steam 
can enter the cylinder when the lever is half-way between its ex- 
treme positions, and that varying amounts of opening of the steam 



Reversing an Engine. 



323 




324 Elements of Mechanism. 

ports, increasing to the maximum value, will occur when the lever 
is pushed over by successive steps. 

ART. 246. Rolling curves have been employed to vary the 
relative angular velocity of two revolving pieces. 

The mathematician Euler appears to have been the first to 
refer to a class of curves which, when caused to turn about fixed 
centres, should communicate motion to each other by rolling 
contact. He deduced the property that the line of contact re- 
mains always in the straight line joining the centres of the curves. 
The subject made no further progress until it was taken up by 
the Rev. H. Holditch, of Cambridge, who investigated the forms of 
several rolling curves and worked out the mathematical theory. 
(See 'Cambridge Phil. Trans.' vol. vii. A.D. 1838.) 

Prop. Where two curved plates, centred upon fixed axes, 
roll together, the point of contact must always lie in the line of 
centres. 

In order to establish this proposition we may reason as 
follows : 

Let two curves have centres at A and C, and suppose that P 
and Q represent two points which will come together when the 
curves move each other by rolling contact. 

Then P describes a circular arc round A as the plate revolves, 
FlG 323 and Q describes a cir- 

cular arc round C. 
Hence P and Q will 
come into contact when- 
ever these circles meet 
each other. Now P and 
Q are only in contact at 
one point, because the 
curves roll, and do not 
slide on each other, and therefore the circular arcs can only meet 
once, that is, they touch each other : but they can only touch in 
the line AC, therefore the point of rolling contact must lie in AC. 
This is equivalent to saying that AP -f CQ = a constant 
quantity. 

Further than this, the curves will have a common tangent at 
the point where they roll upon each other. 




Rolling Curves. 325 

Ex. Two equal ellipses which are centred on opposite foci will 
roll together. 

It is the property of an ellipse that the tangent at any point 
P makes equal angles with the focal distances SP and HP, that is, 
that the angles SPT and HP/ are equal to one another at every 
point of the curve ; and again, that the sum of the lines SP and 
HP is a constant quantity (fig. 324). 

FIG. 324. FIG. 325. 





The two equal ellipses centred upon opposite foci aie repre- 
sented in contact at P (fig. 325). 

Let PT be the tangent to the ellipse A at P, and P/ the tan- 
gent to the ellipse D at P. 

Then SPT = /PH by the property above stated, /. TP/ is a 
straight line, or the curves have a common tangent at the point P, 
also SP + HP = a constant quantity, and the two conditions of 
rolling are fulfilled. 

ART. 247. The theory is simple, and is the following : 

Let SP = r, ASP = 6, HP = r', DHP = ff, the curves AP, 
DP, representing any pair of rolling curves with centres at S and 
H respectively (fig. 325). Then r + / = c. 

///-) ///V 

Also the curves have a common tangent, .*. r= r' . 

dr dr 

Let the differential equation to the first curve be r -- = f(r\ 
then the differential equation to the second curve will be 



whence the relation between 0' and r 1 may be found by inte 
gration. 




326 Elements of Mechanism. 

Ex. There is a curve known as the logarithmic spiral, which 
will roll without sliding upon a second similar and equal spiral. 

o_ 

The equation to the logarithmic spiral is r = aei, and the 
mode of setting out its form is extremely ingenious. 

Draw a number of lines radiating from S, and inclined to SA 
at angles a, 2a, 3a, &c., respectively. 

Then it will be readily seen that, if r\, r 2 , r 3 , &c., are the 
corresponding values of SP, we have the following relations sel 
up, viz. : 

r { r 3 = r 2 2 , r z r^ = r 3 2 , and so on. 

In order to find the lengths of r b r 2 , r 3 , &c., we draw 

straight line S^, inclined 

FlG> 32<5 ' at an angle a to SA, anc 

draw A^r perpendiculai 
to Sjf, xy perpendiculai 
to SA, yz perpendiculai 
to S.r, zv perpendiculai 
to SA, and so on. 
Then SA x S_y = S* 2 , by similar triangles, 

also, Sz; x Sy = Sz 2 , and so on. 

.'. S^, S_y, S^, &c., are the respective values of r\, r 2 , r 3) &c. 
whence the curve can be set out. 

The curve has the property that the tangent at every point i: 
inclined at the same angle to the radius vector at the point COP' 

sidered. 

e 
Taking the equation r = aem, we have 



.'. r - = m, a constant quantity. 
dr 

But r is the tangent of the angle which the curve makes 
dr 

with the radius vector at any point. Let this angle be </>, therefore 
tan </> = a constant, or </> is invariable. 

, dy di) 

Also r -- , = r m, 

dr' dr 

whence the second rolling curve is identical with the first spiral. 



Rolling Ellipses, 



327 




In like manner we could prove that two equal ellipses centred 
upon- opposite foci would roll together. 

248. In practice rolling curves must be provided with 
upon the retreating edge, other- 
wise the driver would leave the follower, 
and the revolution would not be com- 
pleted (fig. 327). 

As is usual in all cases where seg- 
mental wheels are employed, a guide 
must direct the teeth to the exact point 
where they commence to engage each 
other. 

The guide may be dispensed with 
by carrying the teeth all round the 
curves : this construction is usually 
adopted in practice, although, strictly speaking, it destroys the 
rolling action entirely. 

A quick return of the table in small planing machines has been 
effected by the aid of rolling ellipses. 

The table is driven by a crank and connecting rod, and the 
crank exists under the form of a flat circular plate, centred on one 
of the foci, and having a groove radiating from the axis as a line 
of attachment for one end of the connecting rod. As the plate 
may be set in any position upon the elliptical wheel, we propose 
to inquire what will be the effect of a change of direction in the 
groove or crank. 

Let the ellipses have the position shown in fig. 328, S and H 
being the centres of motion, and 
SPHQ being perpendicular to Aa, 
the axis of one ellipse. Draw PR 
perpendicular to Dd, and let the 
ellipse DRdH? be the driver, ro- 
tating as shown. 

While P^/R is rolling upon 
PrtQ, the ellipse Aa makes half a 
revolution ; and while RDP is 
rolling upon QAP, it makes the 
remaining half-revolution. 



FIG. 328. 




328 



Elements of Mechanism. 



Suppose T)d to revolve uniformly, then the times of a half re- 
volution of Aa will be in the same proportion to each other as the 
angle PSR to the angle 360 PSR. The quick half revolution 
occurs when the shaded segments are rolling upon each other. 
If, therefore, the table be made to move in the line HS produced, 
and the crank be placed in a direction perpendicular to Aa, we 
shall obtain the greatest possible difference between the periods of 
advance and return. 

The practical difficulty with rolling wheels exists during that 
part of the revolution where the driver tends to leave the follower, 
and it can only be obviated by making the teeth unusually deep ; 
also the wheels should work in a horizontal plane. 

ART. 249. An instance of rolling curves is exhibited in the 
sketch, and occurred in one of the many attempts made to improve 
the printing press before the invention of Mr. Cowper enabled the 
newspapers to commence a real and vigorous existence. 

The type was placed upon each of the four flat sides of a rect- 
angular prism, to which the wheel B corresponded in shape, and 
the paper was passed on to a platten 
corresponding in form and size with the 
pitch-line of the wheel A. 

The prism and platten being in the 
same relative position as the wheels B 
and A, we can understand that the 
type would be in the act of impressing 
the paper while the convex edge of 
the wheel A rolled upon the flat side 
of B, and that in this way we should 
obtain four impressions for each revo- 
lution of the wheels. 

By this construction, the patentees, 
Messrs. Bacon and Donkin, intended 
to introduce the principle of continuous 
rotation as opposed to the reciprocating 
movement in a common press ; and the 
object of imitating exactly upon the 
wheels A and B the form of the print- 
ing prism and of the platten, was to 



FIG. 329. 




Hookcs Joint. 



329 




ensure that the paper and type should roll upon one another with 
exactly equal velocities at their opposing surfaces, and that no 
slipping or inequality of motion should destroy the sharpness of 
the impression. 

ART. 250. Hooke's Joint is a method of connecting two axes, 
whose directions meet in a point, in 
such a manner that the rotation of 
one axis shall be communicated to 
the other. 

Here AB and CD represent two 
axes whose directions meet in the 
point O ; the extremities of AB and 
CD terminate in two semicircular 
arms which carry a cross, PQSR ; 
the arms of this cross are perfectly 
equal, and the joints P, Q, S, and R permit the necessary free- 
dom of motion. 

As the axis AB revolves, the points P and Q describe a circle 
whose plane is perpendicular to AB, and at the same time the 
points S and R describe another circle whose plane 
is perpendicular to CD. 

These two circles are inclined at the same angle 
as the axes, and are represented in fig. 331 ; thus 
the arm OP starts from P, and moves in the circle 
PP'L, while the arm OR starts from R, and de- 
scribes the circle RR'Q inclined to the former. 

Let OP', OR' be corresponding positions of 
the two arms, then P'R' is constant, but changes 
its inclination at every instant, and as a consequence the relative 
angular velocities of OP' and OR' are continually changing. 

To find the relative angular velocities of the axes AB and CD, 
we proceed as follows : 

Let the circle prq (fig. 332) represent the path of P, ptq being 
the projection upon this circle of the path of R, and suppose a to 
be the angle between AB and CD ; then the dimensions of the 
curve ptq, which vill be an ellipse, can be at once deduced from 
the equation O/ Or cos a." 

Draw Rm perpendicular to Or, then Rm will be the actual 




330 



Elements of Mechanism. 



vertical space through which OR has descended while OP de- 
scribes the angle /K)P. But the path of R is really a circle, and 
only appears to be an ellipse by reason of its being projected upon 



FIG. 332. 





FIG. 334. 




a plane inclined to its own plane. In order, therefore, to estimate 
the actual angular space through which OR has moved, we must 
refer this motion to the circle which R really describes (fig. 333), 
and thus by making R'm' = Rm, we can infer that w'OR' will be 
the angle which OR describes while OP moves through the angle 
/OP. 

But the angle /OP = angle wOR, and hence we can represent 
the motion of both axes upon one diagram by combining the 
ellipse /R^ with the circle prR.'g (fig. 334). This being done, we 
may draw R'RN perpendicular to/O^, and join OR, OR' ; it will 
at once appear that the angles ROr, R'Or are those described in 
the same time by the axes AB and CD. 

Hence the axes AB and CD revolve together, but unequally, 
and the angles which they describe in the same time can always 
be found by construction. 

First draw the circle prq in a plane perpendicular to one axis, 
and having O for its centre, next construct the ellipse whose major 
axis is the diameter pQg equal to POQ, and whose minor axis is 
the product of POQ x the cosine of the angle between the axes. 
Then take OR' any position of OP, draw R'RN perpendicular to 
pQq, join OR' and OR. It now appears that R'Or and ROr will 
represent the angles described by the axes AB and CD in the 
given time. 

Furthermore, OR and OR' coincide when R is at the end of 
an axis of the ellipse /R^, an event which must happen four 



Double Hookes Joint. 



331 



times as the cross goes round once ; and there is therefore this 
curious result, that however unequal may be the rate at which the 
axes are at any time revolving, they will coincide in relative posi- 
tion four times in one revolution. 

The single joint may often be very useful in light machinery 
which is required to be movable, and the parts of which do not 
admit of very accurate adjustment ; but it will be understood 
that the friction, and especially that irregularity which we have 
just proved to exist, would render it necessary to confine the 
angle between the shafts within narrow limits in actual practice. 

ART. 251. Now that the general character of the movement 
is understood, we shall be in a position to comprehend the 
change which is effected by interposing a double joint between 
the axes. 

FIG. 335. 

'D 





i. Take the case where AB and CD are parallel axes con- 
nected by an intermediate piece BC, having a Hooke's joint at 
both the points B and C. 

Conceive that the arms of the crosses at B and C are placed in 
the manner shown in the sketch, or let each vertical arm be con- 
nected with the forks at B and C. 

If AB revolves uniformly, BC will also revolve with a varying 
velocity dependent upon the angle ABC, but the variable velocity 
which BC receives from AB is precisely the same as that which 
it would receive from DC if the latter axis were the driver and 
were to revolve uniformly. 



332 Elements of Mechanism. 

It follows therefore that the motion which AB imparts to CD 
will be a uniform velocity of rotation exactly equal to that 
of AB. 

Hence a double Hooke's joint may be used to communicate 
uniform motion between two parallel axes whose directions nearly 
coincide. 

If, however, the construction were varied and the vertical arm 
PQ of the first cross were connected with E, while the horizontal 
arms, s, r, were connected with F, we should communicate no 
doubt a motion of rotation between the axes, but it would no 
longer be uniform but variable, by reason that we could not return 
by the same course reversed under like conditions. The devi- 
ations from uniform rotation would no longer oppose and correct. 
each other, but they would act together and increase the in- 
equality. This is seen at once upon constructing the diagrams 
which represent the relative rotation between each pair of axes. 

2. Let AB and CD be inclined to each other, and be con- 
nected by the piece BC jointed at B and C, and so placed that 
the angle ABC is equal to the angle BCD. 

As in the former case we must be careful to connect B and C 
with the corresponding arms of the crosses, and we have seen that 
the inequality produced by DC in the motion of CB depends 
both upon the angle BCD and the position of the cross ; it is 
therefore the same whether CD lies in the direction shown, or in 
the dotted line CH parallel to AB. In both cases the angle 
between the axes and the position of the cross will respectively 
coincide. 

But we have seen that when the parallel axes AB and CH are 
connected by joints at B and C in the manner stated, the axes AB 
and CH will rotate with equal uniform velocities, and we conclude, 
therefore, that they will also rotate in a similar manner when 
placed in the position ABCD. 

Hence a double Hooke's joint may be employed to com- 
municate a uniform rotation between two axes inclined at a given 
angle. 

ART. 252. It will be found that well-known propositions of 
Euclid obtain a new significance when applied to movable com- 
binations of the lines of figure. 



Parallel Axes. 



333 



FIG. 336. 




Referring again to the triangle from which we deduced the 
law of motion of the crank and connecting rod : 

i. Let APB represent such a 
triangle, A and B being fixed 
points, and the angle APB being 
a rigid angle. Also, let the sides 
PA, PB, be produced indefinitely 
in order that the dimensions of / c ^^ 

the triangle may vary by the ' 
sliding of PA, PB through the points A and B respectively. 

Let PAB = 0, PBA = 0, 
then + <t> = 180 APB = a constant. 

.*. $9 + cty = o, or ?0 = c<t>, 

whence the angular velocity of PA about A is equal and opposite 
to the angular velocity of PB about B. 

Also since APB is a rigid angle, and since the angles in the 
same segment of a circle are equal to one another, we infer that 
the point P lies always in a circular arc passing through A and B. 

It will simplify the result if we take APB 90, as we can 
then apply the property that the angle in a semicircle is a right 
angle, also AB will in that case be the diameter of the circle 
traced out by the point P. 

In fig. 337 take A and B to represent two fixed axes, and let 
DEFH be a rigid rectangular cross whose arms can slide through 
the points A and B. 

The motion will only be possible so long as the arms of the 
cross have perfect liberty to slide through the points A and B as 
well as to rotate about them. 

Let this be arranged, and join 
AB ; then we have 
PAB + PBA=9o in every position 
of the cross. 

Hence if the angle PAB increase 
by the rotation of PA, the angle 
PBA must diminish equally by the 
rotation of PB, or ED and FH 
must revolve with equal angular 
velocities. 



Fro. 337. 




334 Elements of Mechanism. 

Also P, the angle of the cross, will describe a circle whose 
diameter is AB, and our proposition follows directly from Euclid, 
for if P move on to any point Q, the angles QAP, QBP are 
angles in the same segment of a circle, and are therefore equal 
to each other. 

This movement was put into a practical shape by Mr. Oldham, 
and used in machinery at the Bank of England. The student 
may easily construct a model after the manner of Hooke's Joint, 
when the centre of the cross will be seen to describe a circle 
whose diameter is the perpendicular distance between the axes 
while the arms of the cross slide to and fro through holes in the 
forked arms that spring from the axes and support them. 

2. Bisect AB in C, and take P and C as fixed centres of 
motion. As before, let APB be a right angle, then the rotation of 
AB about C will set up a rotation in both PA and PB, whereby 
each of the latter lines will tend to rotate with half the angular 
velocity of AB. 

In order to make the motion continuous, the lines AP, BP 
must be produced so as to form a rectangular cross, and they 
may be conveniently formed as straight grooves in a plain board 
whose axis passes through P. The bar AB will then be provided 
with pins working in the grooves. 

Describe a circle with centre C and radius equal to CA or 
CB ; draw any fixed diameter A'CB', and join A'P. 

FIG. 33 8. Then it is proved in Euclid that 

the angle at the centre of a circle is 
double the angle at the circumfer- 
ence when both angles stand upon 
the same arc, 

that is angle ACA'=2 angle APA', 
or the angular velocity of CA is 
twice that of the cross. 

As the driver ACB revolves the 
pins A and B will oscillate to and fro 
along their respective grooves and 
will traverse through the centre P. 

ART. 253. The differential worm wheel and tangent screiv is a 
combination- which will be understood without any drawing. 




The Geneva Stop. 335 

Here two worm wheels, differing by one tooth in the number 
which they carry, are placed side by side and close together, so 
as to engage with an endless screw. As the wheels are so very 
nearly alike the endless screw can drive them both at the same 
time, and it is evident that one wheel will turn relatively to the 
other, through the space of the extra tooth, in a complete revo- 
lution, and that a very slow relative motion will thus be set up. 

In this way, if one wheel carries a dial plate, and the other a 
hand, we may obtain the record of a very large number of revolu- 
tions of the tangent screw. 

ART. 254. Where a train of wheels is set in motion by a 
spring enclosed in a barrel it becomes of consequence not to 
overwind the spring. The Geneva stop has been contrived with 
the view of preventing such an occurrence, and will be found in 
all watches which have not a fusee. 

Here a disc A, furnished with one projecting tooth, P, is fixed 
upon the axis of the barrel containing the mainspring, and is 
turned by the key of the watch. 

Another disc, B, shaped as in the drawing, is also fitted to the 
cover of the barrel, and is turned onward in one direction through 
a definite angle every time that the tooth P FIG. 339. 

passes through one of its openings, being 
locked or prevented from moving at other 
times by the action of the convex surface of 
the disc A. 

In this manner each rotation of A will 
advance B through a certain space, and the 
motion will continue until the convex surface 
of A meets the convex portion E, which is 
allowed to remain upon the disc B, in order 
to stop the winding up. 

The winding action having ceased, the discs will return to 
their normal positions as the mechanism runs down. 

Instead of supposing A to make complete revolutions let it 
oscillate to and fro through somewhat more than a right angle ; 
then B will oscillate in like manner and will be held firmly by the 
opposition of the convex to the concave surface except during 
the time that P is moving in the notch. 




336 Elements of Mechanism. 

ART. 255. The Geneva stop has been applied by Sir J. 
Whitworth in his planing machine, in order to give a definite 
vibration to a piece from which the feed motion is derived. 

FIG. 340. 




The drawing shows a lever centred at A, and having a pin P 
at one end. The other end of the lever is weighted at W, the 
object of the weight being to cause the lever to fall over suddenly, 
and with sufficient power to carry the driving belt from one 
working pulley to the next in order. 

A pinion on a shaft terminating at A gears into a rack formed 
on a traversing rod which is moved longitudinally in alternate 
directions by tappets on the table of the machine. The rod, in 
its turn, actuates a bell crank lever which is connected with a fork 
employed for passing the driving belt from one pulley to another. 

As far as this explanation has gone it would appear that the 
weighted lever was designed simply for controlling the driving 
belt, but it will be seen that a portion of a Geneva stop is super- 
added, and this extra piece enables the lever to actuate also the 
feed motion. The axis A of the lever is surrounded by a circular 
plate, corresponding with A in fig. 339, and the piece BED has 
two concave circular cheeks at E and D exactly fitting the plate 
A. Also the pin P works in the open jaws as shown. 

The result is that the lever PA carries BED as it swings 
over, and locks it in either position, whether to the right or the 
left In the diagram the pin P is shown (i) in a vertical posi- 



The Star Wheel. 337 

tion, while in the act of driving BED, and also, (2) after having 
fallen over to the right, at which time BED is securely locked. 

The extreme positions of the weighted lever are shown by 
the dotted lines pa, qb, and it only remains to point out that the 
feed motion is taken from the axis B by means of a grooved 
pulley and a catgut band. This band runs round another pulley 
which has already been described in Art. 132, and is there marked 
as F, and by a comparison with the previous description the 
general arrangement will be readily understood. The locking 
of BED is essential in order to prevent any motion of the cutter, 
before the completion of the cut. 

ART. 256. The star wheel is used in cotton-spinning machi- 
nery, and is analogous to the Geneva stop. 

If the convex portion E were removed, so as 
not to interfere with the rotation of A, we should 
virtually possess a star wheel in the disc B. See 
Art. 254, and fig. 339. 

In that case each rotation of A would advance 
B by the space of one tooth, or we should convert 
a continuous circular motion into one of an inter- 
mittent character. 

The usual form of the star wheel is given in the 
sketch, where the revolving arm encounters and 
carries forward a tooth at each revolution. The 
action is the same as if a wheel with one tooth were to drive 
another with several teeth. 

ART. 257. It is well known to mathematicians that the an- 
gular velocity of a rigid body about an axis may be properly repre- 
sented by a straight line in the direction of that axis, whence it 
follows that angular velocities may be combined according to the 
law which gives us the parallelogram of linear velocities or the 
parallelogram of forces. 

A remarkable illustration of the compounding of angular velo- 
cities has been afforded in the construction of the so-called 
Plimpton or roller skate for use on artificial ice. 

This skate runs like a wagon upon four wheels, but instead 
of the perch-pin being vertical, as in an ordinary wagon, it inclines 
z 




338 



Elements of Mechanism. 



inwards, and dips towards the centre of the skate. Indeed, each 
pair of wheels is provided with an inclined axis, and it will be 
presently seen that the skate will not serve its purpose unless the 
respective axes converge downwards to a point underneath the 
centre of the footboard. 

Everyone is aware of the manner in which curves are described 
by a skater upon ice. For example, when tracing out a circular 
sweep on, say, the right foot, technically distinguished as 'an 
outside edge,' the plan is to keep the leg and body quite straight 
and to lean over a little towards the centre of the circle. 

With a roller skate the same movement of the body produces 
the same result, but in a very different manner. The act of tilting 
over the footboard causes the fore wheels to deviate a little to- 
wards the right, and the hind w.heels to deviate a little towards 
the left, the respective axles of the front and hind pair meeting, 
as they ought to meet, in the centre of the circular path described 
by the skater, and our object is to exhibit a method of construction 
which necessarily produces this result. 




In order to simplify the explanation it will be better to confine 
our attention to the front pair of wheels, and the drawing shows a 
model which may illustrate the movement of the wheels as conse- 
quent upon the tilting of the footboard. 

The right-hand figure gives a perspective view of one half of 
the footboard AB, together with the inclined axis ae, and a pair 
of rollers attached to an axle standing at right angles to a pipe 
or hollow tube, which is threaded upon the immovable inclined 
axis. 

The next sketch shows the footboard resting on its wheels 
near the edge of a horizontal table, and it is apparent that if the 



Roller Skates. 339 

skate were pushed forward a little it would advance in a line per- 
pendicular to the edge of the table. 

Now raise one edge of the footboard without in any way 
altering the direction in which its central dotted line points, and 
let it be noted that the direction of that line is at right angles to 
the edge of the table. 

In the drawing the footboard is shown as tilted through an 
angle xvy, and the immediate result is that the axle of the rollers 
turns to the right in the manner indicated by the arrow, and that 
if the skate were pushed forward it would immediately move in a 
curved line pointing to the left hand. 

It is extremely easy to construct the model, and the move- 
ment may then be studied with advantage. 

In applying general reasoning we say that there are three 
axes of rotation before us, and it will be better to take the case 
where the board in the model is held in a horizontal position 
parallel to the plane of the table. In exhibiting the model it is 
more convenient to hold it in this way, but the drawing is clearer 
when the board is allowed to drop with one end on the table. 

The three axes of rotation are : 

1. A horizontal axis through the horizontal footboard AB. 

2. The inclined axis ae. 

3. A vertical axis through a. 

Here two simultaneous rotations about the vertical and hori- 
zontal axes may give a resultant rotation about the inclined axis, 
just as horizontal and vertical forces acting on a point have an in 
clined resultant. 

But again, in the case of forces, the combination of the in- 
clined resultant with the horizontal component would give the 
other vertical component ; so here, the combination of the rotation 
about the horizontal axis AB with another rotation about the 
inclined axis ae, gives a rotation about an imaginary vertical axis, 
viz., that passing through a, and hence the axle of the rollers does 
in effect rotate about a vertical axis, just as if it wefe the axle of 
an ordinary carriage provided with a vertical perch-pin. Such a 
rotation causes the rollers to deviate on one side of the normal 
direction. 

The specification of Mr. Plimpton's patent, granted to A. V. 



34 Elements of Mechanism. 

Newton and numbered 2190 of the series for 1865, contains draw- 
ings of the skate, and shows the foot stand running upon four 
wheels, two on each side of the roller axle. The respective axles 
are supported in frames capable of turning to a small extent 
limited by stops, and are directed downwards to a point half-way 
between the heel and toe of the skate. These ledges perform the 
function of inclined axes. The invention is described as relating 
to an improvement in attaching rollers to the foot stand of a skate, 
whereby the rollers are made to turn by the rocking of the foot 
stand so as to cause the skates to run in a curved line either to 
the right or left. 

It has been stated by experts on the subject that the arrange- 
ment of two inclined axes for causing the roller axles to converge 
towards the centre of curvature of the path of the skater was 
a completely new invention at the date of the patent, and that 
no machine existed at that time in which a like motion had been 
arrived at in so simple a manner. 



APPENDIX 



INTRODUCTORY TO REULEAUX'S SYSTEM OF 
TEACHING MECHANISM 



In the following Appendix the writer proposes to make some 
observations upon a distinct method of analysing combinations of 
mechanism, as originated a few years ago by Professor Reuleaux, 
of Berlin, in a work entitled the ' Kinematics of Machinery,' which 
has been translated into English by Mr. Kennedy. 

It is scarcely possible to preface the inquiry by a general state- 
ment of the method adopted, inasmuch as the technical terms 
employed would be unintelligible without explanation, and should 
be carefully defined in the first instance. Accordingly, we com- 
mence with some general definitions, and shall allow the subject- 
matter to develop itself gradually. 

Art. L Def. : When a body is constrained to move in a definite 
manner by means of an envelope the combination is termed a pair. 
The envelope forms one element of the pair, and the enclosed 
body forms the other element. 

Such pairs are of two kinds, viz. higher pairs and lower pairs, 
the combinations which are formed being distinguished by the 
general terms, higher pairing and loiver pairing. 

It is essential at the outset to appreciate and understand this 
distinction, which runs through the whole subject- matter. 

Def. : When a body and its envelope are in surface contact, 
and so constructed that every point of the body is constrained to 
describe a straight line, all such straight lines being parallel to 
each other, the two pieces form a sliding pair. 

Such a pair is described and shown at p. 17 of this book. 

Def. : When a body and its envelope are in surface contact, 



342 Elements of Meclianism. 

and so constructed that every point in the body is constrained 
to describe a circle, all such circles being in parallel planes and 
having a common axis, the two pieces form a turning pair. 

Such a pair is described and shown (ante) at pp. 17 and 18. 

Def. : When a body and its envelope are in surface contact, 
and so constructed that every point in the body is constrained to 
describe a screw thread of uniform pitch and having a common 
axis, the two pieces form a screw pair. 

An example is afforded by any ordinary and well-made screw 
and nut. 

It appears that there are only three kinds of lower pairs, namely, 
those before mentioned, which provide for the following move- 
ments : 

1. Simple straight-line motion. 

2. Circular motion. 

3. The combination of rectilinear and circular motion in a 
given fixed ratio. 

Def. : Where one element completely surrounds the other so 
that no motion is possible except that which the combination is 
intended to produce, the pair is said to be dosed. 

In order to make these definitions clear, take the case of a 
horizontal shaft revolving in an ordinary circular bearing made in 
two halves and bolted together. 

If there are shoulders upon the shaft just outside the bearing, 
so as to prevent endlong motion, the pair will be complete, or 
will be a closed pair. 

If the bolts be taken out of the bearing and the upper half 
lifted off, it may be that the shaft will go on rotating just as before, 
being held in position by its weight or otherwise. The motion 
is, therefore, that of a turning pair, and there is artificial closure, 
because the desired movement is arrived at but the combination 
no longer forms a closed pair. It is, in fact, evident that the 
shaft may be lifted off its bearing while the rotation is going on, 
which would make an end of the turning pair. 

This result is shortly expressed by saying that the pair is not 
a closed pair. 

It will presently be necessary to refer to other analogous use 
of the word closure. 



Pairs and Chains. 343 

2. There are, as we have stated, only three lower pairs, but 
the number of higher pairs cannot be estimated. 

There is higher pairing when two toothed wheels are brought 
into gear, inasmuch as there is definite constrained motion, but 
with line and not surface contact between two teeth, and it is 
hardly necessary to remark that one tooth is not an envelope of 
the other. Also, there is both rolling and sliding at the parts 
where the teeth are in contact. Reuleaux gives, as a fundamental 
example of higher pairing, a rectangular block sliding in a groove 
of varying curvature, where the motion may be constrained and 
definite, but where surface contact cannot take place. 

It is a property of all pairs, whether of the higher or lower 
kind, that, if one element be fixed, a definite motion can be given 
to the other element. It is also a fundamental property of a pair 
of elements that, if one element be fixed, every motion of the 
second element, except the particular motion required, is prevented. 

3. The term chain, or kinematic chain, is applied to any com- 
bination of pairs wherein motion of the several parts is admissible. 

But a chain is made up of links, and accordingly, when two 
elements of different pairs in a chain are connected together, the 
elements so connected form a link in the chain. While the pairs 
are loose and unconnected there is no chain, but, as soon as the 
links are formed all through, the chain is ready for the final step 
which makes it an operative instrument. 

The final step is the fixing of one link, whereby the chain is 
supported and can exhibit its properties. 

Def. : If a chain be so constructed that, when one link is 
fixed, each other link can only accept one definite and determinate 
motion, the chain is said to be closed. 

Def. : A closed kinematic chain, in which one link is fixed, 
is called a mechanism. 

Def. : A mechanism, when set in motion by a mechanical force 
applied to one of its links, is called a machine. 

From what has preceded, it is clear that the order of arrange- 
ment of the links in a chain is material, and that changing the 
order in which the pairs are arranged may alter the properties of 
the chain in the communication of motion. 

Since a mechanism is derived by fixing a link in a chain it 



344 Elements of Mechanism 

follows that the number of mechanisms which can be formed 
from a chain is equal to the number of links in the chain. 
Whether the mechanisms so formed are the same or different will 
depend upon other considerations. 

4. The word ' chain ' having a technical meaning, and being 
used in a sense quite different from that ordinarily attributed to 
it, we propose to examine certain well-known combinations from 
the new point of view. 

For this purpose we shall commence with instances of lower 
pairing, by taking chains formed with four pairs of simple elements. 

For simplicity, let s represent a sliding pair, and let T re- 
present a turning pair. Also let the selected chains be : 

F.G. I. 




where the first diagram represents a simple combination of four 
turning pairs, after which one or two sliding pairs replace the 
corresponding turning pairs. 

It appears that the four-bar motion described (ante) page no 
is a combination of four turning pairs, the elemei/s of which are 
united, so as to constitute four links, namely, c p, P Q, Q B, B c. 
When one link (such as c B) is fixed, the arrangement exhibits the 
properties of a chain or mechanism, and the necessary calculations 
as to the relative motions of the 
separate parts have already been 
worked out. Also, since one T is 
the same as another, the combina- 

p ^^ \\ tion will only furnish one chain and 

one distinct mechanism, and, although 
there are practical differences with 
( o & reference to the fact that sometimes 

B Q oscillates, and at other times per- 
forms complete revolutions while c P revolves, there is only one 




Four-Bar Motion. 



345 



general proposition proved at pages in and 112, which gives the 
relative velocities of c P and B Q, or of c p and P Q. 

5. Taking, however, the second combination as made up of 
the pairs x T, T s, it appears that we have one chain from which 
we can deduce two distinct mechanisms, namely, those given by 
fixing the positions of the links derived from 

T s and T T. 

The chain itself is called a slider-crank chain. 
Also, we restrict the investigation to cases where the direction 
of sliding passes always through the centre of one turning pair. 

6. Taking the direct acting horizontal engine, with a crank 
and connecting rod, as shown in the diagram, there is manifestly 
a chain made up of three turning pairs and one sliding pair. The 
turning pairs are marked c, p, Q in the drawing, and the sliding 
pair is the block at Q moving in the slides H K, L M. Inasmuch 
as the position of c relatively to the slides is fixed by attaching 
the bearings of the shaft at c, together with the slides, to the 



FIG. 3. 




framework of the engine, it is clear that the fixed link is derived 
from T s by uniting one element of T with one element of s. 

The first mechanism deducible from the slider- crank chain is 
therefore to be found in any ordinary locomotive engine. 

7. Another mechanism is obtained by deriving the fixed link 
from T T, that is, by uniting an element of one turning pair with 
an element of another turning pair. 

This is done in the Whitworth shaping machine described and 
figured (ante} at pages 102 and 103. 



346 



Elements of Mechanism. 



It appears that in the Whitworth machine a small shaft is in- 
serted into a bearing B, bored out in the main shaft, which is 
sufficiently large for the purpose. Hence the centres of two 
turning pairs are fixed, namely, the centre B, and the main shaft 
which carries the wheel F, and a new mechanism results, but the 
chain itself differs in no respect from that which exists in the 
ordinary crank and connecting rod. 




That the chain cannot vary is seen from an inspection of 
Fig. i, which shows that wherever s is inserted the three T's 
must always follow in order, and hence that the relative motion 
of the separate links will be the same. The slider-crank chain 
is therefore always one and the same chain, but two distinct 
mechanisms can be derived from it, namely, those enumerated 
above. 

8. It is part of our subject to consider the manner in which by 
varying the details of construction any given mechanism may 
appear in a disguised form. 

FIG. 5. 




The slider-crank chain being taken as an example, we shall 
deal first with the mechanism where T s is the fixed link, as shown 
in the diagram. 



Slider- Crank Cliain. 



347 



1. The crank CP may be replaced by a pin p attached to the 
face of a plate riding upon a shaft c, as in the Whitworth shaping 
machine. 

2. The pin at p may be enlarged so as to embrace the 
centre c, and the envelope of the pin may take the form of a 
hoop attached by a rod to the sliding block Q. This is the 




construction of an ordinary eccentric and rod as employed for 
actuating a slide valve (see page 51). 

If we regard the eccentric in this manner, it becomes un- 
necessary to prove that the combination is an equivalent for the 
crank and connecting rod, inasmuch as either arrangement gives 
one and the same mechanism. 

3. The next step is to enlarge the pin at Q so as to make it 
embrace the shaft c with its eccentric circle, and to surround 

FIG. 7- 




this pin by a hoop furnished with sliding guides which pass through 
bearings at A, B, as in the diagram. 

It is apparent that when the shaft c revolves carrying the 
eccentric circle with it, the enlarged pin R will oscillate about 
the centre Q, and we shall have the point Q reciprocating in the 



348 Elements of Mechanism. 

line C Q, just as in the ordinary crank and connecting-rod arrange- 
ment. 

4. If in the normal diagram the pin at Q be united to the 
envelope of the slider at Q we have the fixed link s T as it stood 
originally. 

But P Q will now oscillate, and c must travel to and fro in 
the line Q c in order that P c may perform complete revolutions 
about P. 

The reciprocation of c has been taken advantage of in Stannah's 
pump. 

5. We pass on to other variations of the mechanism where 
T T is a fixed link. 




The Whitworth shaping machine being the fundamental in- 
stance of this mechanism, it appears 

That if in the diagram of the Whitworth machine the crank c P 
be made less than CQ, the slotted bar being QP, there will be 
oscillation of Q P instead of rotation. 

This is the mechanism of a well-known quick-return shaping 
machine, as described at p. 104, which may take the form of an 
oscillating cylinder engine, by increasing the distance c Q, and 
by converting the slotted link into a cylinder oscillating upon 
trunnions at Q. 

6. The mechanism of the oscillating engine is also apparent 
from the normal diagram. 

If a fixed link connects the turning pairs at P and Q, and the 
sliding guides H K, L M be converted into a cylinder oscillating on 
trunnions at E, and provided with a piston Q, it becomes apparent 
that the arrangement is the same as that of the shaping machine 
just referred to, but in a disguised form. 



Double Slider- Crank Chain. 



349 



9. The remaining combination in Fig. i is that of two turning 
and two sliding pairs, which is technically known as 
The double slider-crank chain. 

Here the order may be s s T T or s T s T at pleasure, and the 
result is that we can form two distinct chains. 

These chains will be expressed in a diagram as shown. 



FIG. 9. 





Fig. (i) will give three distinct mechanisms namely, those 
obtained by fixing the links derived respectively from 
ss, TT, ST; 

and the following examples of these several mechanisms are sug- 
gested. 

i. The mechanism derived from fixing ss is to be found in 
the elliptic compasses. The construction is the following : 




In two grooves A B, c D, at right angles to each other, are fitted 
two rectangular blocks bored for the reception of the pins E, F. 

A rod E F p, carrying a pencil at p, is attached to the pins, 
whereby the pencil P is competent to describe an ellipse on the 



350 



Elements of Mechanism. 



plane A B c D. The sliding pairs are the blocks and grooves, while 
the turning pairs are the pins E, F, with their envelopes. 

2. The mechanism derived from fixing XT is to be found in 
Oldham's coupling, where two fixed shafts, whose directions are 
parallel and lie close together, are to be united by a coupling so 
that either may drive the other. 

Here each shaft terminates in a disc with a rectilinear slot or 
groove, and a flat-faced block having a corresponding rib or pro- 
jection on each side engages with and couples together the t'vo 
shafts, the directions of the ribs when projected on a plane parallel 
to either face of the block being at right angles to each other. 
The block will slide up and down in the grooves as the shafts 
revolve, and will communicate the necessary rotation. 

3. An example of the mechanism where s T supplies the fixed 
link is to be found in the bullet machine described (ante) at 
P a e 55- Here the centre of motion of the shaft c is fixed, and 
the guides in which the shaft ABDE slides are also fixed. 




The first turning pair is the shaft c with its bearings, the second 
turning pair is the eccentric disc P working in the square block. 

The sliding pairs are the block and frame, together with the 
reciprocating shaft and its bearings. 

Also, the fixed link is obtained by connecting the bearing of 
the shaft c with the guides in which the shaft ABDE reciprocates. 

The mechanism is that of a crank with an infinite link giving 
a simple harmonic motion. 

Fig. (2) will give only one mechanism- namely, that derived 
by fixing the position of 



Double Slider-Crank CJiain. 



351 



The mechanism so obtained is not commonly met with, but 
may be seen in the annexed form of steering gear by J. Rapson 
(patent No. 8214 of 1839). 

By this device the turning moment on the tiller remains con- 
stant in every position thereof. To effect this object a block 
Q R slides in guides D E, F H and carries a pin which runs through 
another sliding block p, which can move along the tiller c B as 
shown, the axis of the tiller being at c. The direction of the 
pull on the tiller ropes is shown by the arrows, and it is obvious 




that the pull on the tiller caused by turning the steering wheel 
remains constant at any inclination of the tiller to the dotted line. 

Here the combination is s T s' T', where 

s is the sliding pair Q R with its guides, 

T is the turning pair at c, 

s' is the sliding pair made up of the block and c B. 

T' is the turning pair at the pin p. 

Also the fixed link is ST. 

Note. It is important that one fundamental property of a chain 
should be clearly understood, namely, that in the same chain, 
whatever may be the mechanisms formed from it, the velocity- 
ratio of like parts is always the same. Thus, for example, when 
the student has worked out the velocity ratios for the Whitworth 
shaping machine, he can apply his results to the oscillating engine 

10. In the three elementary pairs mentioned in Art. I, the 
reciprocal restraint of the elements is complete. That is why the 
term ' closed ' is introduced. 



35 2 Elements of Mechanism. 

If the reciprocal restraint be incomplete, some kind of closure 
will become necessary. 

Thus, if two spur wheels, as A and K, be brought into gear, they 
form, as we have said, a higher pair, but unless they are sup- 
ported and held at the right distance, they will not work properly. 
The reciprocal restraint which is indispensable is here supposed 
to be obtained by carefully holding the wheels and allowing them 
to turn on their axes without separating. It is obvious that this 
could not be done effectively, and the above method, which would 
be an example of force closure, is not attempted in practice, but 
in place thereof it is common to mount the wheels A and B upon 
pins at their centres and to support the pins in a bar or frame 
which is rigidly supported. It will be apparent that two simple 
turning pairs have thus been introduced, giving a chain with two 




turning pairs and one higher pair. This is an example of pair 
closure. 

A couple of spur wheels in gear, when supported on a fixed 
frame and turning freely, is to be regarded therefore as a chain with 
three pairs. 

If the position of the wheel A be fixed, and the bar carrying B 
be allowed to rotate about the axis of A we arrive at an epicyclic 
train, giving the so-called sun and planet motion. 

But the chain in the case of the spur wheels with fixed axes 
is precisely the same as in the epicyclic train, and it follows that 
a proposition to the effect that the relative motion of the wheel B 
to the wheel A in the epicyclic train is the same as its relative 
motion to the wheel A in the train with fixed axes is a direct 
consequence of the fact that the chain is the same in both 
cases. 

This example is worth consideration because it leads to the 



Chain Closure. 353 

inference that there may be other things to be thought of beyond 
the grouping of pairs into chains and mechanisms. 

It is well known to mechanics, and is in part apparent from 
the results set forth in the chapter on Aggregate Motion, that an 
epicyclic train exhibits properties of practical value and is com- 
petent to furnish results which would never have been arrived at 
by the use of trains of wheels with fixed axes. 

The method now before us gives but small assistance in inter- 
preting the peculiarities of an epicyclic train, or in showing to a 
student what may be done with it. 

11. Hitherto the possible defect of closure in a pair has been 
pointed out, and it remains to explain what is meant by defect 
of chain closure. 

An instance of defect of closure in a chain is to be found in 
the jointed parallelogram c P B Q. 

If c P be the driver, the position of c B being fixed, it is apparent 
that when P is crossing the line of centres, namely c B, the arm Q B 
may continue to rotate in the same direction as c P, or may begin 
to move backward. Either motion is equally possible, the chain 
is unclosed, but QB may be constrained to rotate in the same 
direction as c P by introducing a new chain CP'Q'B as already 
explained (ante p. 115). 

FIG. 14. 
C B 



\ A 




This is therefore an example of the closure of one chain by 
the addition of another chain, and such a device is technically 
known as chain closure. 

In constructing models to exhibit any special movement, it is 
a common thing to employ some kind of closure for getting over 
difficulties which may arise. Thus (ante) at page 141 it is shown 
that when c P = P Q in the slider crank chain there is a failure of 
motion when Q arrives at c, and Mr. Booth, in a patent, No. 9824 



354 Elements of Mechanism. 

of 1843, has shown how to deal with it. The method referred to 
is a case of pair closure. Thus Q p is produced to R, making 
p R = c P, and a pin fastened at R is caused to engage with a 
forked or pronged opening in the right position on c D, so as to 
introduce a supplemental turning pair just as Q arrives at c, whereby 
the motion is carried on, and the throw of Q is four times the 
length of the crank c P, or twice as great as in the ordinary move- 
ment. 

If the student has grasped the meaning which underlies these 
short notes, he will comprehend the leading idea which dominates 
throughout the German method of studying mechanism. By 
regarding combinations as chains of moving parts, he may be 
enabled to consolidate and group together a number of possible 
mechanisms which in practice assume forms apparently unlike, 
but in one sense identical, and after the system has been more 
fully worked out and applied it may become of value and im- 
portance as an additional means of generalising results. 



INDEX, 



AGG 

A GGREGATE motion. 215-65 
* Alarum clock, 68 
Annular wheel, 21 
Arbor, axis, axle, 17 
Archimedean drill, 166 
Axis, instantaneous, 32 



BALANCE wheel of watch, 293 
Bell crank levers, 40-41 
Belts or bands, 25 

transfer of motion by, 25 

how kept on pulleys, 26 

open or crossed, 26 

with axes at right angles, 28 
Bevel wheels, 22 

teeth of, 1 88 

skew, 312 . 

Blower, by Root, 313 

Bobbin motion, by Houldsworth, 233 

model to illustrate, 235 

application of, 238 

Bodmer, drilling machine by, 252 
Boring machine, 253 

feed motion of, 254 

use of epicyclic train in, 255 

Brace, ratchet, 152 



CALLIPERS, use of, 275 

^ Cam, definition of, 68 

use of, in conversion of motion, 69 

analysis of curve in simple cases, 69, 70 

as a heart wheel, 71 

for imitating handwriting, 72 

in sewing machine, 73 

altered form of, 74 

in striking mechanism of a clock, 74 

in a lever punching machine, 75 

in printing machine, 75 

in a rifling machine, 76 



CRA 

Cam described on cylinder, 77, 78 

example of, 79 

other examples, from printing ma- 
chinery, 80-2 

double, 82 

expansion, 83 

for multiplied oscillations, 84, 128 

used in carpet weaving, 128 
Cartwright's Cordelier, 244 
Centrode, meaning of term, 32 
Change wheels, 208 
Chinese windlass, 220 
Chronometer escapement, 2935 
Chronometric governor, by Siemens, 316 
Circles, angular velocity ratio of, in roll- 
ing, 23 

Circular, into reciprocating motion, 42 
Circular motion, of a point, 8, 9 

transfer of, 15 

relation of angles described in, 16 

transmission of, 15, 16, 20 

converted into reciprocating, 42 

same by wheclwork, 85 

Circumduction, motion of, 1 16 
Clock train, 201 
Clutch, 91 

Combination of motions of translation 
and rotation, 33 

of two and three spur wheels, 86 

example of same, in screwing ma- 
chine, 87 

in planing machine, 88 

of two cranks and link, numerous ex- 
amples, 110-27 

Cones, rolling of, 23 
Conical pulleys, 200 
Copying machinery, examples of, 92, 195, 

206, 292 

Cordelier, by Cartwright, 244 
Counting wheels, 284 
Crab, lilting, 203 



356 



Index. 



CRA 



Crane, wheel-work of, 204 
Crank, and connecting rod, 42 

analysis of motion of, 45-8 

throw of, 45 

contrivance for doubling throw of, 141 

same by wheelwork, 216 

expanding, 263 

variable, rotation of, 100 
Cranks, two, with link, 110-27 

one oscillating, 113 

applied in wool combing, 118 

and in ventilating machine, 119 

and in sewing machine, 120 

and in shearing machine, 121 

and in Stanhope levers, 123 

multiple rotating, 115-16 

example of, 247 
Crown wheel, 21 

escapement, 59 

Curvature, circle of, 171 
Cycloid, definition of, 180 
Cylindrical gauges, 279 



"pVEAD points, 149 
4-/ Difference gauges, 280 
Differential pulley, for carriages, by Sax 
ton, 217 

screw, 218 

pulley, by Weston, 221 

motion, for cotton spinning, 236-40 
Disc and roller, 309 

applied for continuous indicator, 

310 

Drilling machine, principle of, 249 
Drill spindle, motion of, 249 

example of, 250 

by Bodmer, 252 

by Sir J. Whitworth, 256 

Driver, meaning of term, 20 



ECCENTRIC circle, properties of, 49 
Eccentric, throw of, 50 

construction in steam engine, 51 

example of, 52 

use of in drilling machines, 53 

equivalent for crank and connecting 
rod, 50 

same for crank and infinite link, 54 

example of, 55 

End measure, standard bars, 283 

conversion into line measure, 275 

machine for, by Sir J. Whitworth, 

276 

Epicloid, definition of, 173 
Epicyclic tram, 222 
theory of, 224 



Epicyclic train, for straight line motion, 229 

model to illustrate, 230 

for astronomical models, 231 

with bevel wheels, 232 

in spinning machinery, 233 

for slow motion, 240 

compared with ordinary train, 242 

in rope-making, 245 

in Cordelier, 243-8 

Equation clock, 243 
Escapement, simple form, 58 

crown wheel, 59 

anchor, 62 

with recoil, 62 

dead beat, 66 

examples of recoil, 68 

pin wheel, 68 

chronometer, 293-5 

detached lever, 296 

horizontal, by Graham, 297 

FEATHERING paddle wheel, 117 
* Feed motion, 155 

of rifling machine, 155-6 

by Sir J. Whitworth, 157 

silent, 159 

by Worssam, 160 

of boring machine, 254 

same by epicyclic train, 255 

Ferguson's paradox, 227 
Follower, meaning of term, io 
Foot-second, definition of, 5 
Four-bar motion, 110-15 
Fusee, 298 

theory of, 299 

flat spiral, used in cotton spinning ma- 
chinery, 301 

other examples, 302 

principle of, winding-on motion, 303 

going, by Harrison, 305 



GAUGES, standard cylindrical, 279 
difference, 280 
Gear, meaning of term, 22 
Geneva stop, 335 

applied in planing machine, 336 

Glass-grinding machine, 262 
Governor, of steam engine, by Watt, 314 
chronometric, by Siemens, 316 
Graham's cylinder "escapement, 297 
Grasshopper engine, parallel motion, 142 
Guide pulleys, 30-1 



i H 



ARMONIC motion, simple, 9 
amplitude of, 10 



Index. 



357 



HAR 

Harmonic motion, period of, 10 

phase of, i to 

analysis of, n 

composition of, 12 

model to illustrate, 14, 42 

application of, 39 

Harrison's going fusee, 305 
Heart wheel, 71 

modified form of, 74 

Hooke's joint, 329-32 
Horizontal escapement, 297 
Hunting cog, 214 
Hypocycloid, definition of, 173 
'property of, 176 



T OLE wheel, 194 

* Indicator, invented by Watt, 257 

by McXaught, 258 

by Richards, 260 

diagram, 258 

application of parallel motion in, 261 

continuous, by Ashton and Storey, 310 
Instantaneous axis, 32 

Intermittent motion, example of, 57 

theory of same, 58 

by segmental wheels, 108-9 

Involute teeth, 183-5 



TACK, lifting, 38 
J Joint, toggle, 125 

applied in printing machinery, 126 
in carpet weaving, 127 

for multiplied vibrations, 128 

Hooke's, theory of, 329 

double Hooke's, 331 



J EYLESS watch, 306-7 

LATHE, Blanchard, 195 
screw cutting, 206 
Lazy tongs, 215 
Lemie'le's ventilator, 119 
Lever, bell crank, 40 

nipping, 158 

escapement, 296 
Levers, Stanhope, 122 

theory of same, 124-5 

of Lagarousse, 162 

modified form of same, 163 
Line and end measure, 275 
Link-work, meaning of term, no 
Link and two cranks, 1 10 

theory of same, 111-13 

one crank oscillating, 113-14 



Link and two cranks, both cranks rota- 
ting, 115 

applied in wool combing, 118 

and for shuttle motion in sew- 
ing machine, 129 

also in machine for cutting 

metals, 121 

also in Stanhope levers, 122 

Link motion, by Stephenson, 320 

by Gooch, 322 

Logarithmic spiral, setting out curve, 326 

MACHINE for shaping the naves of 
wheels, 291 

shaping, with quick return, 103 

another form, 104 

for drilling, 250 

for measuring. 276 
Mandril of a lathe, 18 
Mangle rack, 105 

for Cowper's printing machine, 106 

double, with segmental wheel, 108 

single, with segmental wheels, 108 

Mangle wheel, 105 

with quick return, 106 

Marlborough wheel, 196 
Masked ratchet, use of, 291 
Measuring bars, preparation of, 273 
Measuring machine, by Sir J . Whitworth, 
276 

millionth, 281 

Mitre wheels, 23 
Motion of rotation, 33 

of translation, 33 

of hour hand, 202 



AJIPPING lever, 158 

L^ Numbering machine, 286-291 



OSCILLATION, centre of, 61 
model to illustrate, 62 

multiplied, 127 

Oval chuck, theory of, 263-5 



PADDLE wheel, feathering, 117 
Pairs, elementary, 17 

examples of, 17 

cylindric, four parallel, no 
Pallets, 58 

Pantograph, 137 

in parallel motion of beam engine, 137 
Paradox, Ferguson's, 227 

Parallel axes, transfer of motion between, 
323-4 



358 



Index. 



Parallel motion, by Watt, 129 

theory of, 130-4 

of beam engine, 137 

of marine engine, 138 

application of, 139 

in grashopper engines, 142 

in compound engines, 146 

- in Richards' indicator, 261 
Paul, meaning of term, 150 

action of, 151 

Pauls, of unequal length, 153 

application of, 154 
Peaucellier's straight line motion, 143 
Pendulum, simple, bo 

law of oscillation, 60 

centre of oscillation of rigid, 61 

model to illustrate same, 62 

mechanical action of, 65 
Pin wheels, theory of, 179 
Pinion, meaning of term, 22 

lantern, 179 
Pitch, diametral, 168 

circular, 169 

Photometer, Wheatstone's, 177 
Planing machine, reversing motion, 89 
feed motion, 157 

Plimpton's roller skate, 338 
Point, motion of a, 4 
circular motion of, 9 

harmonic motion of, 9 

Power, telodynamic transmission of, 29 
Pulley, convex rim of, 26 

single movable, 219 
Pulleys, fast and loose, 27 

with inclined axes, 28 

guide, 30 

speed, 196 

theory of same, 197 

applied in lathe, 199 

conical, 200 



RACK and pinion, forms of teeth, 182 
teeth derived from involute 

of circle, 188 
Rack, mangle, 105 
-- double, 108, 216 
Ratchet wheel, 150 

for driving in alternate directions, 

I 5 I 
used in lifting jack, 152 

with click and hook, 162 
masked, 163-5 

compared with lifting pump, 167 

Ratchet brace, 152 

Reciprocating into circular motion, 148 

examples, 166 
Rectangular bars, 273 



Reversing motion, by spur wheels, 85 

with quick return, 86, 88 

example of, 87 

by disc wheels, 90 

by bevel wheels, 91 

by clutch, 91 

adopted by Sir J. Whitworth, 92 

example in rifling machine, 94 

by pulleys and belts, 96 

by slit bar and crank, 97 

theory of same, 97-101 

by double eccentric and link motion. 



j Roberts's winding-on motion, 303 

I Rolling curves, theory of, 324 
examples, 325 

for quick return, 327 

in printing machinery, 328 

Root's blower, 313 
Rope, twist of, 245 

model to illustrate same, 247 

extra twist, apparatus for, 248 



CAXTON'S differential pulley, 217 
J Scraping tool, 269 
Screw surface, definition of, 34 
Screw, pitch of, 34 

right or left handed, 34, 207 

single or double threaded, 35 
Screw threads, 34 

mechanical properties of, 34-6 

uniform system of, by Sir J. Whit- 
worth, 37 

Screw coupling, 219 

Screw cutting, theory of, 205 

lathe for, 206-8 

Screw and worm wheel, used as rack and 
pinion, 256 

Sector, use of, 170 

Segmental wheels, 108, 109 

Shaping machine, with quick return, by 
Sir J. Whitworth, 102 

for locomotive wheels, 104 

Siemens's chronometric governor, 316 

Silent feed, 159 

Similar curves, 135 

Skate, Plimpton's, 338 

Skew bevels, 312 

Slit bar motion, 97 

theory of, 97-101 

Slow motion, by epicyclic train, 240 

Snail, 308 

Speed pulleys, 196-7 

Spiral logarithmic, 326 

Standard gauges, 279 

Standards of length, 283 

Stanhope levers, 123 



Index. 



359 



Star wheel, 337 

Step wheels, 312 

Straight line motion, by Scptt Russell, 140 

theory of, 140 

exact, by Peaucellier. 143 

multiple, 146-7 

by epicyclic train, 229 

Sun and planet wheels, 226 
Surface plate, 268 

appearance of, 270 

method of preparing, 271 

adhesion of, 272 



Swash plate, 55 
theory of, 56 



TEETH, involute, theory of, 183 
action of same, 185 

contact of, in wheelwork, 213 
Teeth of wheels, theory of, 172 

general solution, 174 

first case, 177 

with radial flanks, 178 

second case, 178 

with involute curves, 183-5 

general considerations, 185-7 

lor bevel wheels, 188 

Throw of crank, 45 

doubled, 141 

Toggle joint, 125 
Tooth, root or flank, 21 

point of, 21 

pitch of, 21 

Trains of wheels for given purposes, 209-14 
Trains, epicyclic, 223-48 
True plane, meaning of term, 267 
Truth of surface, importance of, 266 
Twist of a rope, 247 



\7 ELOCITY, how measured, 5 
how represented, 6 

angular, 18 

measure of, 18 
Velocities, parallelogram of, 7 

triangle of, 8 



WOR 



Velocities, diagram of, in harmonic mo- 
tion, 13 

Velocity ratio, 20 

with parallel axes, 24 

with inclined axes, 24 

between crank pin and piston in 

direct-acting engine, 47 

diagram of same, 48 

in cam motion, 77 

also for oscillating engine, 99 

between crank and slit bar, 99-100 

in four-bar motion, in 

Vibrations, multiplied, 128* 



, keyless, 306-7 
* v Watts' indicator, 257 

parallel motion, 129 
Weston's differential pulley, 2*1 
Wheel, toothed, 20 

pitch circle of, 20 

spur, 21 

crown, 21 

annular, 21 

bevel, 21 

face, 21 

worm, 37 

segmental, 108-5 

Marlborough, 196 

purchase, 216 

with racks for doubling throw of crank, 
216 

Wheels, segmental, 108-9 

in trains, 190 

examples of, 192-4 

in clock train, 201 

for motion of hour hand, 202 

in lifting crab, 203 

in crane, 204 

step, 312 

Whitworth, Sir J., measuring machine, 

refer to Chapter VIII. 
Windlass, Chinese, 220 
Worm wheel, 37 
Worm barrel, 8r 
with movable switch, 82 



INDEX TO APPENDIX. 



DULLET machine, 350 



f^HAIN, or kinematic chain, 343 
^ of four turning pairs, 344 

of three turning and one sliding 
pair, 345 

slider crank, 345 
- property of, 351 

of three pairs, 352 
Closure, meaning of term, 342 

defect of, 353 

example of chain, 353 

same of pair, 353 
Compasses, elliptic, 349 
Coupling, Oldham's, 350 

TROUBLE slider crank chain, 349 

elementary mechanisms, 349, 
350, 351 



"PCCENTRIC, example of slider 
" crank chain, 347 



Epicyclic train, 352 

is chain of three pairs, 



353 



"CORCE closure, 352 

Four-bar motion, 344 
only one mechanism, 344 

T INK of chain, 343 



MACHINE, 343 

Mechanism, definition of, 343 



QSCILLATING engine, 348 
^ belongs to slider cr; 



chain, 348 



crank 



TDAIR, meaning of the term, 341 

sliding, turning, screw, 341, 
342 

closed, 342 

closure, 352 
Pairing, lower, 344 

higher, 352 

QUICK RETURN shaping machine, 
348 

"DAPSON'S steering gear, 351 

CLIDER crank chain, 345 

*^ elementary mechanisms, 

345, 246 

other forms, 347, 348 
without rods, 347 

Stannah's pump, 34-8 

WHITWORTH shaping machine, 
348 

what chain it belongs to, 348 

same mechanism as oscillating 
engine, 348 



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